{"appState":{"pageLoadApiCallsStatus":true},"categoryState":{"relatedCategories":{"headers":{"timestamp":"2023-02-01T16:01:11+00:00"},"categoryId":34229,"data":{"title":"Calculation & Analysis","slug":"calculation-analysis","image":{"src":null,"width":0,"height":0},"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"parentCategory":{"categoryId":34226,"title":"Accounting","slug":"accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"}},"childCategories":[],"description":"Whether you're trying to figure out the difference between covariance and correlation or breaking out a regression equation, our articles have the info you need.","relatedArticles":{"self":"https://dummies-api.dummies.com/v2/articles?category=34229&offset=0&size=5"},"hasArticle":true,"hasBook":true,"articleCount":216,"bookCount":3},"_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"relatedCategoriesLoadedStatus":"success"},"listState":{"list":{"count":10,"total":216,"items":[{"headers":{"creationTime":"2016-03-26T08:13:24+00:00","modifiedTime":"2022-10-06T17:13:01+00:00","timestamp":"2022-10-06T18:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Business Statistics: Test the Estimated Regression Equation","strippedTitle":"business statistics: test the estimated regression equation","slug":"test-the-estimated-regression-equation-using-the-coefficient-of-determination-r2","canonicalUrl":"","seo":{"metaDescription":"You can check whether a regression equation makes sense by using the coefficient of determination, also known as R2 (R squared).","noIndex":0,"noFollow":0},"content":"After you estimate the population regression line, you can check whether the regression equation makes sense by using the coefficient of determination, also known as <i>R</i><sup>2</sup> (<i>R</i> squared). This is used as a measure of how well the regression equation actually describes the relationship between the dependent variable (<i>Y</i>) and the independent variable (<i>X</i>).\r\n\r\nIt may be the case that there is no real relationship between the dependent and independent variables; simple regression generates results even if this is the case. It is, therefore, important to subject the regression results to some key tests that enable you to determine if the results are reliable.\r\n\r\nThe coefficient of determination, <i>R</i><sup>2</sup>, is a statistical measure that shows the proportion of <i>variation</i> explained by the estimated regression line. Variation refers to the sum of the squared differences between the values of <i>Y</i> and the mean value of <i>Y</i>, expressed mathematically as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460867.image0.png\" alt=\"image0.png\" width=\"91\" height=\"53\" />\r\n\r\n<i>R</i><sup>2</sup> always takes on a value between 0 and 1. The closer <i>R</i><sup>2</sup> is to 1, the better the estimated regression equation fits or explains the relationship between <i>X</i> and <i>Y</i>.\r\n\r\nThe expression\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460868.image1.png\" alt=\"image1.png\" width=\"91\" height=\"53\" />\r\n\r\nis also known as the <i>total sum of squares</i> (TSS).\r\n\r\nThis sum can be divided into the following two categories:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Explained sum of squares (ESS):</b> Also known as the <i>explained variation</i>, the ESS is the portion of total variation that measures how well the regression equation explains the relationship between <i>X</i> and <i>Y</i>.</p>\r\n<p class=\"child-para\">You compute the ESS with the formula</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460869.image2.png\" alt=\"image2.png\" width=\"143\" height=\"53\" /></li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Residual sum of squares (RSS):</b> This expression is also known as <i>unexplained variation</i> and is the portion of total variation that measures discrepancies (errors) between the actual values of <i>Y</i> and those estimated by the regression equation.</p>\r\n<p class=\"child-para\">You compute the RSS with the formula</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460870.image3.png\" alt=\"image3.png\" width=\"147\" height=\"53\" /></li>\r\n</ul>\r\nThe smaller the value of RSS relative to ESS, the better the regression line fits or explains the relationship between the dependent and independent variable.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Total sum of squares (TSS):</b></p>\r\n<p class=\"child-para\">The sum of RSS and ESS equals TSS.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460871.image4.png\" alt=\"image4.png\" width=\"297\" height=\"53\" />\r\n<p class=\"child-para\"><i>R</i><sup>2</sup> is the ratio of explained sum of squares (ESS) to total sum of squares (TSS):</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460872.image5.png\" alt=\"image5.png\" width=\"83\" height=\"47\" />\r\n<p class=\"child-para\">You can also use this formula:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460873.image6.png\" alt=\"image6.png\" width=\"105\" height=\"47\" />\r\n<p class=\"child-para\">Based on the definition of <i>R</i><sup>2</sup>, its value can never be negative. Also, <i>R</i><sup>2</sup> can't be greater than 1, so</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460874.image7.png\" alt=\"image7.png\" width=\"77\" height=\"24\" /></li>\r\n</ul>\r\nWith simple regression analysis, <i>R</i><sup>2</sup> equals the square of the correlation between <i>X</i> and <i>Y</i>.\r\n\r\nThe coefficient of determination is used as a measure of how well a regression line explains the relationship between a dependent variable (<i>Y</i>) and an independent variable (<i>X</i>). The closer the coefficient of determination is to 1, the more closely the regression line fits the sample data.\r\n\r\nThe coefficient of determination is computed from the sums of squares. These calculations are summarized in the following table.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460875.image8.png\" alt=\"image8.png\" width=\"447\" height=\"319\" />\r\n\r\nTo compute ESS, you subtract the mean value of <i>Y</i> from each of the estimated values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460876.image9.png\" alt=\"image9.png\" width=\"205\" height=\"53\" />\r\n\r\nTo compute RSS, you subtract the estimated value of <i>Y</i> from each of the actual values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460877.image10.png\" alt=\"image10.png\" width=\"197\" height=\"53\" />\r\n\r\nTo compute TSS, you subtract the mean value of <i>Y</i> from each of the actual values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460878.image11.png\" alt=\"image11.png\" width=\"191\" height=\"53\" />\r\n\r\nAlternatively, you can simply add ESS and RSS to obtain TSS:\r\n\r\nTSS = ESS + RSS = 0.54 + 0.14 = 0.68\r\n\r\nThe coefficient of determination (<i>R</i><sup>2</sup>) is the ratio of ESS to TSS:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460879.image12.png\" alt=\"image12.png\" width=\"204\" height=\"47\" />\r\n\r\nThis shows that 79.41 percent of the variation in <i>Y</i> is explained by variation in <i>X</i>. Because the coefficient of determination can't exceed 100 percent, a value of 79.41 indicates that the regression line closely matches the actual sample data.","description":"After you estimate the population regression line, you can check whether the regression equation makes sense by using the coefficient of determination, also known as <i>R</i><sup>2</sup> (<i>R</i> squared). This is used as a measure of how well the regression equation actually describes the relationship between the dependent variable (<i>Y</i>) and the independent variable (<i>X</i>).\r\n\r\nIt may be the case that there is no real relationship between the dependent and independent variables; simple regression generates results even if this is the case. It is, therefore, important to subject the regression results to some key tests that enable you to determine if the results are reliable.\r\n\r\nThe coefficient of determination, <i>R</i><sup>2</sup>, is a statistical measure that shows the proportion of <i>variation</i> explained by the estimated regression line. Variation refers to the sum of the squared differences between the values of <i>Y</i> and the mean value of <i>Y</i>, expressed mathematically as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460867.image0.png\" alt=\"image0.png\" width=\"91\" height=\"53\" />\r\n\r\n<i>R</i><sup>2</sup> always takes on a value between 0 and 1. The closer <i>R</i><sup>2</sup> is to 1, the better the estimated regression equation fits or explains the relationship between <i>X</i> and <i>Y</i>.\r\n\r\nThe expression\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460868.image1.png\" alt=\"image1.png\" width=\"91\" height=\"53\" />\r\n\r\nis also known as the <i>total sum of squares</i> (TSS).\r\n\r\nThis sum can be divided into the following two categories:\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Explained sum of squares (ESS):</b> Also known as the <i>explained variation</i>, the ESS is the portion of total variation that measures how well the regression equation explains the relationship between <i>X</i> and <i>Y</i>.</p>\r\n<p class=\"child-para\">You compute the ESS with the formula</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460869.image2.png\" alt=\"image2.png\" width=\"143\" height=\"53\" /></li>\r\n \t<li>\r\n<p class=\"first-para\"><b>Residual sum of squares (RSS):</b> This expression is also known as <i>unexplained variation</i> and is the portion of total variation that measures discrepancies (errors) between the actual values of <i>Y</i> and those estimated by the regression equation.</p>\r\n<p class=\"child-para\">You compute the RSS with the formula</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460870.image3.png\" alt=\"image3.png\" width=\"147\" height=\"53\" /></li>\r\n</ul>\r\nThe smaller the value of RSS relative to ESS, the better the regression line fits or explains the relationship between the dependent and independent variable.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Total sum of squares (TSS):</b></p>\r\n<p class=\"child-para\">The sum of RSS and ESS equals TSS.</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460871.image4.png\" alt=\"image4.png\" width=\"297\" height=\"53\" />\r\n<p class=\"child-para\"><i>R</i><sup>2</sup> is the ratio of explained sum of squares (ESS) to total sum of squares (TSS):</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460872.image5.png\" alt=\"image5.png\" width=\"83\" height=\"47\" />\r\n<p class=\"child-para\">You can also use this formula:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460873.image6.png\" alt=\"image6.png\" width=\"105\" height=\"47\" />\r\n<p class=\"child-para\">Based on the definition of <i>R</i><sup>2</sup>, its value can never be negative. Also, <i>R</i><sup>2</sup> can't be greater than 1, so</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460874.image7.png\" alt=\"image7.png\" width=\"77\" height=\"24\" /></li>\r\n</ul>\r\nWith simple regression analysis, <i>R</i><sup>2</sup> equals the square of the correlation between <i>X</i> and <i>Y</i>.\r\n\r\nThe coefficient of determination is used as a measure of how well a regression line explains the relationship between a dependent variable (<i>Y</i>) and an independent variable (<i>X</i>). The closer the coefficient of determination is to 1, the more closely the regression line fits the sample data.\r\n\r\nThe coefficient of determination is computed from the sums of squares. These calculations are summarized in the following table.\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460875.image8.png\" alt=\"image8.png\" width=\"447\" height=\"319\" />\r\n\r\nTo compute ESS, you subtract the mean value of <i>Y</i> from each of the estimated values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460876.image9.png\" alt=\"image9.png\" width=\"205\" height=\"53\" />\r\n\r\nTo compute RSS, you subtract the estimated value of <i>Y</i> from each of the actual values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460877.image10.png\" alt=\"image10.png\" width=\"197\" height=\"53\" />\r\n\r\nTo compute TSS, you subtract the mean value of <i>Y</i> from each of the actual values of <i>Y</i>; each term is squared and then added together:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460878.image11.png\" alt=\"image11.png\" width=\"191\" height=\"53\" />\r\n\r\nAlternatively, you can simply add ESS and RSS to obtain TSS:\r\n\r\nTSS = ESS + RSS = 0.54 + 0.14 = 0.68\r\n\r\nThe coefficient of determination (<i>R</i><sup>2</sup>) is the ratio of ESS to TSS:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460879.image12.png\" alt=\"image12.png\" width=\"204\" height=\"47\" />\r\n\r\nThis shows that 79.41 percent of the variation in <i>Y</i> is explained by variation in <i>X</i>. Because the coefficient of determination can't exceed 100 percent, a value of 79.41 indicates that the regression line closely matches the actual sample data.","blurb":"","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":207822,"title":"Business Statistics For Dummies Cheat Sheet","slug":"business-statistics-for-dummies-cheat-sheet","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/207822"}},{"articleId":162083,"title":"How Businesses Use Regression Analysis Statistics","slug":"how-businesses-use-regression-analysis-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162083"}},{"articleId":162074,"title":"Random Variables and Probability Distributions in Business Statistics","slug":"random-variables-and-probability-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162074"}},{"articleId":162073,"title":"Explore Hypothesis Testing in Business Statistics","slug":"explore-hypothesis-testing-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162073"}},{"articleId":162066,"title":"3 Ways to Describe Populations and Samples in Business Statistics","slug":"3-ways-to-describe-populations-and-samples-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162066"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282040,"slug":"business-statistics-for-dummies","isbn":"9781118630693","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118630696-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/business-statistics-for-dummies-cover-9781118630693-203x255.jpg","width":203,"height":255},"title":"Business Statistics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9080\">Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-633f17df596c3\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-633f17df59dd3\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-10-06T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":145936},{"headers":{"creationTime":"2016-03-26T08:14:22+00:00","modifiedTime":"2022-09-22T16:12:14+00:00","timestamp":"2022-09-22T18:01:03+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Convert a Sampling Distribution Using the Central Limit Theorem","strippedTitle":"how to convert a sampling distribution using the central limit theorem","slug":"how-to-convert-a-sampling-distribution-to-a-standard-normal-random-variable-using-the-central-limit-theorem","canonicalUrl":"","seo":{"metaDescription":"Learn how to convert a sampling distribution to a standard normal random variable using the central limit theorem in business statistics.","noIndex":0,"noFollow":0},"content":"You can use the Central Limit Theorem to convert a sampling distribution to a standard normal random variable. Based on the Central Limit Theorem, if you draw samples from a population that is greater than or equal to 30, then the sample mean is a normally distributed random variable. To determine probabilities for the sample mean\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460170.image0.png\" alt=\"image0.png\" width=\"35\" height=\"25\" />\r\n\r\nthe standard normal tables requires you to convert\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460171.image1.png\" alt=\"image1.png\" width=\"20\" height=\"21\" />\r\n\r\nto a standard normal random variable.\r\n<p class=\"Remember\">The standard normal distribution is the special case where the mean</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460172.image2.png\" alt=\"image2.png\" width=\"27\" height=\"23\" />\r\n\r\nequals 0, and the standard deviation\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460173.image3.png\" alt=\"image3.png\" width=\"28\" height=\"23\" />\r\n\r\nequals 1.\r\n\r\nFor any normally distributed random variable <i>X</i> with a mean\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460174.image4.png\" alt=\"image4.png\" width=\"16\" height=\"19\" />\r\n\r\nand a standard deviation\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460175.image5.png\" alt=\"image5.png\" width=\"20\" height=\"17\" />\r\n\r\nyou find the corresponding standard normal random variable (<i>Z</i>) with the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460176.image6.png\" alt=\"image6.png\" width=\"80\" height=\"47\" />\r\n\r\nFor the sampling distribution of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460177.image7.png\" alt=\"image7.png\" width=\"23\" height=\"24\" />\r\n\r\nthe corresponding equation is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460178.image8.png\" alt=\"image8.png\" width=\"92\" height=\"52\" />\r\n\r\nAs an example, say that there are 10,000 stocks trading each day on a regional stock exchange. It's known from historical experience that the returns to these stocks have a mean value of 10 percent per year, and a standard deviation of 20 percent per year.\r\n\r\nAn investor chooses to buy a random selection of 100 of these stocks for his portfolio. What's the probability that the mean rate of return among these 100 stocks is greater than 8 percent?\r\n\r\nThe investor's portfolio can be thought of as a sample of stocks chosen from the population of stocks trading on the regional exchange. The first step to finding this probability is to compute the moments of the sampling distribution.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Compute the mean: </b></p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460179.image9.png\" alt=\"image9.png\" width=\"112\" height=\"27\" /></li>\r\n</ul>\r\nThe mean of the sampling distribution equals the population mean.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Determine the standard error:</b> This calculation is a little trickier because the standard error depends on the size of the sample relative to the size of the population. In this case, the sample size (<i>n</i>) is 100, while the population size (<i>N</i>) is 10,000. So you first have to compute the sample size relative to the population size, like so:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460180.image10.png\" alt=\"image10.png\" width=\"227\" height=\"25\" />\r\n<p class=\"child-para\">Because 1 percent is less than 5 percent, you don't use the finite population correction factor to compute the standard error. Note that in this case, the value of the finite population correction factor is:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460181.image11.png\" alt=\"image11.png\" width=\"323\" height=\"53\" /></li>\r\n</ul>\r\nBecause this value is so close to 1, using the finite population correction factor in this case would have little or no impact on the resulting probabilities.\r\n\r\nAnd because the finite population correction factor isn't needed in this case, the standard error is computed as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460182.image12.png\" alt=\"image12.png\" width=\"91\" height=\"197\" />\r\n\r\nTo determine the probability that the sample mean is greater than 8 percent, you must now convert the sample mean into a standard normal random variable using the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460183.image13.png\" alt=\"image13.png\" width=\"92\" height=\"52\" />\r\n\r\nTo compute the probability that the sample mean is greater than 8 percent, you apply the previous formula as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460184.image14.png\" alt=\"image14.png\" width=\"252\" height=\"55\" />\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460185.image15.png\" alt=\"image15.png\" width=\"197\" height=\"27\" />\r\n\r\nthese values are substituted into the previous expression as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460186.image16.png\" alt=\"image16.png\" width=\"423\" height=\"55\" />\r\n\r\nYou can calculate this probability by using the properties of the standard normal distribution along with a standard normal table such as this one.\r\n<table><caption>Standard Normal Table — Negative Values</caption>\r\n<tbody>\r\n<tr>\r\n<th><b><i>Z</i></b></th>\r\n<th>0.00</th>\r\n<th>0.01</th>\r\n<th>0.02</th>\r\n<th>0.03</th>\r\n</tr>\r\n<tr>\r\n<td>–1.3</td>\r\n<td>0.0968</td>\r\n<td>0.0951</td>\r\n<td>0.0934</td>\r\n<td>0.0918</td>\r\n</tr>\r\n<tr>\r\n<td>–1.2</td>\r\n<td>0.1151</td>\r\n<td>0.1131</td>\r\n<td>0.1112</td>\r\n<td>0.1093</td>\r\n</tr>\r\n<tr>\r\n<td>–1.1</td>\r\n<td>0.1357</td>\r\n<td>0.1335</td>\r\n<td>0.1314</td>\r\n<td>0.1292</td>\r\n</tr>\r\n<tr>\r\n<td>–1.0</td>\r\n<td>0.1587</td>\r\n<td>0.1562</td>\r\n<td>0.1539</td>\r\n<td>0.1515</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nThe table shows the probability that a standard normal random variable (designated <i>Z</i>) is <i>less than or equal to</i> a specific value. For example, you can write the probability that\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460187.image17.png\" alt=\"image17.png\" width=\"77\" height=\"19\" />\r\n\r\n(one standard deviation below the mean) as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460188.image18.png\" alt=\"image18.png\" width=\"108\" height=\"31\" />\r\n\r\nYou find the probability from the table with these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Locate the first digit before and after the decimal point (–1.0) in the first (<i>Z</i>) column.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the second digit after the decimal point (0.00) in the second (0.00) column.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">See where the row and column intersect to find the probability:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460189.image19.png\" alt=\"image19.png\" width=\"176\" height=\"31\" /></li>\r\n</ol>\r\nBecause you're actually looking for the probability that <i>Z</i> is greater than or equal to –1, one more step is required.\r\n\r\nDue to the symmetry of the standard normal distribution, the probability that <i>Z</i> is greater than or equal to a negative value equals one minus the probability that <i>Z</i> is less than or equal to the same negative value.\r\n\r\nFor example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460190.image20.png\" alt=\"image20.png\" width=\"224\" height=\"31\" />\r\n\r\nThis is because\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460191.image21.png\" alt=\"image21.png\" width=\"188\" height=\"19\" />\r\n\r\nare <i>complementary</i> events. This means that <i>Z</i> must either be greater than or equal to –2 or less than or equal to –2. Therefore,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460192.image22.png\" alt=\"image22.png\" width=\"224\" height=\"31\" />\r\n\r\nThis is true because the occurrence of one of these events is <i>certain</i>, and the probability of a certain event is 1.\r\n\r\nAfter algebraically rewriting this equation, you end up with the following result:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460193.image23.png\" alt=\"image23.png\" width=\"224\" height=\"31\" />\r\n\r\nFor the portfolio example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460194.image24.png\" alt=\"image24.png\" width=\"256\" height=\"57\" />\r\n\r\nThe result shows that there's an 84.13 percent chance that the investor's portfolio will have a mean return greater than 8 percent.","description":"You can use the Central Limit Theorem to convert a sampling distribution to a standard normal random variable. Based on the Central Limit Theorem, if you draw samples from a population that is greater than or equal to 30, then the sample mean is a normally distributed random variable. To determine probabilities for the sample mean\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460170.image0.png\" alt=\"image0.png\" width=\"35\" height=\"25\" />\r\n\r\nthe standard normal tables requires you to convert\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460171.image1.png\" alt=\"image1.png\" width=\"20\" height=\"21\" />\r\n\r\nto a standard normal random variable.\r\n<p class=\"Remember\">The standard normal distribution is the special case where the mean</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460172.image2.png\" alt=\"image2.png\" width=\"27\" height=\"23\" />\r\n\r\nequals 0, and the standard deviation\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460173.image3.png\" alt=\"image3.png\" width=\"28\" height=\"23\" />\r\n\r\nequals 1.\r\n\r\nFor any normally distributed random variable <i>X</i> with a mean\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460174.image4.png\" alt=\"image4.png\" width=\"16\" height=\"19\" />\r\n\r\nand a standard deviation\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460175.image5.png\" alt=\"image5.png\" width=\"20\" height=\"17\" />\r\n\r\nyou find the corresponding standard normal random variable (<i>Z</i>) with the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460176.image6.png\" alt=\"image6.png\" width=\"80\" height=\"47\" />\r\n\r\nFor the sampling distribution of\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460177.image7.png\" alt=\"image7.png\" width=\"23\" height=\"24\" />\r\n\r\nthe corresponding equation is\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460178.image8.png\" alt=\"image8.png\" width=\"92\" height=\"52\" />\r\n\r\nAs an example, say that there are 10,000 stocks trading each day on a regional stock exchange. It's known from historical experience that the returns to these stocks have a mean value of 10 percent per year, and a standard deviation of 20 percent per year.\r\n\r\nAn investor chooses to buy a random selection of 100 of these stocks for his portfolio. What's the probability that the mean rate of return among these 100 stocks is greater than 8 percent?\r\n\r\nThe investor's portfolio can be thought of as a sample of stocks chosen from the population of stocks trading on the regional exchange. The first step to finding this probability is to compute the moments of the sampling distribution.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Compute the mean: </b></p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460179.image9.png\" alt=\"image9.png\" width=\"112\" height=\"27\" /></li>\r\n</ul>\r\nThe mean of the sampling distribution equals the population mean.\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\"><b>Determine the standard error:</b> This calculation is a little trickier because the standard error depends on the size of the sample relative to the size of the population. In this case, the sample size (<i>n</i>) is 100, while the population size (<i>N</i>) is 10,000. So you first have to compute the sample size relative to the population size, like so:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460180.image10.png\" alt=\"image10.png\" width=\"227\" height=\"25\" />\r\n<p class=\"child-para\">Because 1 percent is less than 5 percent, you don't use the finite population correction factor to compute the standard error. Note that in this case, the value of the finite population correction factor is:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460181.image11.png\" alt=\"image11.png\" width=\"323\" height=\"53\" /></li>\r\n</ul>\r\nBecause this value is so close to 1, using the finite population correction factor in this case would have little or no impact on the resulting probabilities.\r\n\r\nAnd because the finite population correction factor isn't needed in this case, the standard error is computed as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460182.image12.png\" alt=\"image12.png\" width=\"91\" height=\"197\" />\r\n\r\nTo determine the probability that the sample mean is greater than 8 percent, you must now convert the sample mean into a standard normal random variable using the following equation:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460183.image13.png\" alt=\"image13.png\" width=\"92\" height=\"52\" />\r\n\r\nTo compute the probability that the sample mean is greater than 8 percent, you apply the previous formula as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460184.image14.png\" alt=\"image14.png\" width=\"252\" height=\"55\" />\r\n\r\nBecause\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460185.image15.png\" alt=\"image15.png\" width=\"197\" height=\"27\" />\r\n\r\nthese values are substituted into the previous expression as follows:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460186.image16.png\" alt=\"image16.png\" width=\"423\" height=\"55\" />\r\n\r\nYou can calculate this probability by using the properties of the standard normal distribution along with a standard normal table such as this one.\r\n<table><caption>Standard Normal Table — Negative Values</caption>\r\n<tbody>\r\n<tr>\r\n<th><b><i>Z</i></b></th>\r\n<th>0.00</th>\r\n<th>0.01</th>\r\n<th>0.02</th>\r\n<th>0.03</th>\r\n</tr>\r\n<tr>\r\n<td>–1.3</td>\r\n<td>0.0968</td>\r\n<td>0.0951</td>\r\n<td>0.0934</td>\r\n<td>0.0918</td>\r\n</tr>\r\n<tr>\r\n<td>–1.2</td>\r\n<td>0.1151</td>\r\n<td>0.1131</td>\r\n<td>0.1112</td>\r\n<td>0.1093</td>\r\n</tr>\r\n<tr>\r\n<td>–1.1</td>\r\n<td>0.1357</td>\r\n<td>0.1335</td>\r\n<td>0.1314</td>\r\n<td>0.1292</td>\r\n</tr>\r\n<tr>\r\n<td>–1.0</td>\r\n<td>0.1587</td>\r\n<td>0.1562</td>\r\n<td>0.1539</td>\r\n<td>0.1515</td>\r\n</tr>\r\n</tbody>\r\n</table>\r\nThe table shows the probability that a standard normal random variable (designated <i>Z</i>) is <i>less than or equal to</i> a specific value. For example, you can write the probability that\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460187.image17.png\" alt=\"image17.png\" width=\"77\" height=\"19\" />\r\n\r\n(one standard deviation below the mean) as\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460188.image18.png\" alt=\"image18.png\" width=\"108\" height=\"31\" />\r\n\r\nYou find the probability from the table with these steps:\r\n<ol class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">Locate the first digit before and after the decimal point (–1.0) in the first (<i>Z</i>) column.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Find the second digit after the decimal point (0.00) in the second (0.00) column.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">See where the row and column intersect to find the probability:</p>\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460189.image19.png\" alt=\"image19.png\" width=\"176\" height=\"31\" /></li>\r\n</ol>\r\nBecause you're actually looking for the probability that <i>Z</i> is greater than or equal to –1, one more step is required.\r\n\r\nDue to the symmetry of the standard normal distribution, the probability that <i>Z</i> is greater than or equal to a negative value equals one minus the probability that <i>Z</i> is less than or equal to the same negative value.\r\n\r\nFor example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460190.image20.png\" alt=\"image20.png\" width=\"224\" height=\"31\" />\r\n\r\nThis is because\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460191.image21.png\" alt=\"image21.png\" width=\"188\" height=\"19\" />\r\n\r\nare <i>complementary</i> events. This means that <i>Z</i> must either be greater than or equal to –2 or less than or equal to –2. Therefore,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460192.image22.png\" alt=\"image22.png\" width=\"224\" height=\"31\" />\r\n\r\nThis is true because the occurrence of one of these events is <i>certain</i>, and the probability of a certain event is 1.\r\n\r\nAfter algebraically rewriting this equation, you end up with the following result:\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460193.image23.png\" alt=\"image23.png\" width=\"224\" height=\"31\" />\r\n\r\nFor the portfolio example,\r\n\r\n<img src=\"https://www.dummies.com/wp-content/uploads/460194.image24.png\" alt=\"image24.png\" width=\"256\" height=\"57\" />\r\n\r\nThe result shows that there's an 84.13 percent chance that the investor's portfolio will have a mean return greater than 8 percent.","blurb":"","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. 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Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-632ca2df3290a\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-632ca2df3320d\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-09-22T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":146051},{"headers":{"creationTime":"2018-06-02T03:01:41+00:00","modifiedTime":"2022-03-08T15:34:59+00:00","timestamp":"2022-09-14T18:19:21+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Finite Math For Dummies Cheat Sheet","strippedTitle":"finite math for dummies cheat sheet","slug":"finite-math-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"Keep these numbers, notations, distributions, and listings right at hand when performing computations found in Finite Math topics.","noIndex":0,"noFollow":0},"content":"When performing the many types of computations found in Finite Math topics, it’s helpful to have some numbers, notations, distributions, and listings right at hand.","description":"When performing the many types of computations found in Finite Math topics, it’s helpful to have some numbers, notations, distributions, and listings right at hand.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221b29a8744\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221b29a8fd4\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":252743,"title":"Pascal’s Triangle","slug":"pascals-triangle","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/252743"}},{"articleId":252748,"title":"Binomial Distributions","slug":"finite-math-binomial-distributions","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/252748"}},{"articleId":252753,"title":"Matrix Notation","slug":"matrix-notation","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/252753"}},{"articleId":252758,"title":"Factorial","slug":"factorial","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/252758"}}],"content":[{"title":"Pascal's triangle","thumb":null,"image":null,"content":"<p>When performing computations in problems involving probability and statistics, it’s often helpful to have the binomial coefficients found in Pascal’s triangle. These numbers are the results of finding combinations of <em>n</em> things taken <em>k</em> at a time. For quick reference, the first ten rows of the triangle are shown.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252745\" src=\"https://www.dummies.com/wp-content/uploads/finitemath-pascals.jpg\" alt=\"finitemath-pascals\" width=\"433\" height=\"200\" /></p>\n"},{"title":"Binomial distributions","thumb":null,"image":null,"content":"<p>A common situation when doing probability problems is having to determine the patterns of heads-and-tails, boys-and-girls, true-or-false arrangements. When there are two choices, there are 2<em><sup>n</sup></em> ways that they can occur. The following refer to Heads-or-Tails in a coin-flipping situation but can be adapted to any binomial arrangement.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252749\" src=\"https://www.dummies.com/wp-content/uploads/finitemath-one-toss.jpg\" alt=\"finitemath-one-toss\" width=\"535\" height=\"989\" /></p>\n"},{"title":"Matrix notation","thumb":null,"image":null,"content":"<p>Matrices are rectangular arrangements of elements. The dimension of a matrix is given with <em>m</em> × <em>n</em> where <em>m</em> is the number of rows and <em>n </em>is the number of columns. The elements are identified with subscripts giving the row, <em>j</em>, and column,<em> k</em>, shown as <em>a<sub>jk</sub></em> for the elements of a matrix A.</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252754\" src=\"https://www.dummies.com/wp-content/uploads/Matrix-A.jpg\" alt=\"Matrix-A\" width=\"535\" height=\"335\" /></p>\n<p>When multiplying the matrices, the number of rows in the first matrix has to equal the number of columns in the second. Given matrices A and B where A has dimension 2 × 3 and B has the dimension3 × 2, the resulting matrices are found as follows:</p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-252755\" src=\"https://www.dummies.com/wp-content/uploads/Matrix-B.jpg\" alt=\"Matrix-B\" width=\"535\" height=\"191\" /></p>\n"},{"title":"Factorial","thumb":null,"image":null,"content":"<p>The factorial operation says to multiply the designated number by every positive integer smaller than that number.</p>\n<p><em>n</em>! = <em>n</em> (<em>n</em> – 1) (<em>n</em> – 2)   3 2 1</p>\n<p>When using the operation in the formulas for the number of permutations or combinations of <em>n</em> things taken <em>k</em> at a time, factorial values need to be inserted into the numerator and denominator of the fraction. The first sixteen factorial values are given here. And, by definition, 0! = 1.</p>\n<table>\n<tbody>\n<tr>\n<td width=\"53\"><em>n</em></td>\n<td width=\"108\"><em>n</em>!</td>\n<td width=\"60\"><em>n</em></td>\n<td width=\"168\"><em>n</em>!</td>\n</tr>\n<tr>\n<td width=\"53\">1</td>\n<td width=\"108\">1</td>\n<td width=\"60\">9</td>\n<td width=\"168\">362,880</td>\n</tr>\n<tr>\n<td width=\"53\">2</td>\n<td width=\"108\">2</td>\n<td width=\"60\">10</td>\n<td width=\"168\">3,628,800</td>\n</tr>\n<tr>\n<td width=\"53\">3</td>\n<td width=\"108\">6</td>\n<td width=\"60\">11</td>\n<td width=\"168\">39,916,800</td>\n</tr>\n<tr>\n<td width=\"53\">4</td>\n<td width=\"108\">24</td>\n<td width=\"60\">12</td>\n<td width=\"168\">479,001,600</td>\n</tr>\n<tr>\n<td width=\"53\">5</td>\n<td width=\"108\">120</td>\n<td width=\"60\">13</td>\n<td width=\"168\">6,227,020,800</td>\n</tr>\n<tr>\n<td width=\"53\">6</td>\n<td width=\"108\">720</td>\n<td width=\"60\">14</td>\n<td width=\"168\">87,178,291,200</td>\n</tr>\n<tr>\n<td width=\"53\">7</td>\n<td width=\"108\">5,040</td>\n<td width=\"60\">15</td>\n<td width=\"168\">1,307,674,368,000</td>\n</tr>\n<tr>\n<td width=\"53\">8</td>\n<td width=\"108\">40,320</td>\n<td width=\"60\">16</td>\n<td width=\"168\">20,922,789,888,000</td>\n</tr>\n</tbody>\n</table>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-03-08T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":252761},{"headers":{"creationTime":"2016-03-27T16:48:32+00:00","modifiedTime":"2022-02-16T20:29:06+00:00","timestamp":"2022-09-14T18:19:08+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Reading Financial Reports For Dummies Cheat Sheet","strippedTitle":"reading financial reports for dummies cheat sheet","slug":"reading-financial-reports-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet includes essentials of reading financial reports, including annual reports, SEC reports, profitability ratios, and more.","noIndex":0,"noFollow":0},"content":"If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts.\r\n\r\nCombining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow.\r\n\r\nKeep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.","description":"If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts.\r\n\r\nCombining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow.\r\n\r\nKeep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.","blurb":"","authors":[{"authorId":8974,"name":"Lita Epstein","slug":"lita-epstein","description":" <b>Lita Epstein,</b> who earned her MBA from Emory University&#8217;s Goizueta Business School, enjoys helping people develop good financial, investing and tax-planning skills.<br /> While getting her MBA, Lita worked as a teaching assistant for the financial accounting department and ran the accounting lab. After completing her MBA, she managed finances for a small nonprofit organization and for the facilities management section of a large medical clinic.<br /> She designs and teaches online courses on topics such as investing for retirement, getting ready for tax time and finance and investing for women. She&#8217;s written over 20 books including <i>Reading Financial Reports For Dummies</i> and <i>Trading For Dummies.<br /> </i>Lita was the content director for a financial services Web site, MostChoice.com, and managed the Web site, Investing for Women. As a Congressional press secretary, Lita gained firsthand knowledge about how to work within and around the Federal bureaucracy, which gives her great insight into how government programs work. In the past, Lita has been a daily newspaper reporter, magazine editor, and fundraiser for the international activities of former President Jimmy Carter through The Carter Center.","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8974"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":159960,"title":"10 Mobile Apps for Financial Report Readers","slug":"10-mobile-apps-for-financial-report-readers","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159960"}},{"articleId":159957,"title":"How Internal Financial Reports Differ from the Public Ones","slug":"how-internal-financial-reports-differ-from-the-public-ones","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159957"}},{"articleId":159958,"title":"Key Tools on the Web for Analyzing Numbers","slug":"key-tools-on-the-web-for-analyzing-numbers","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159958"}},{"articleId":159959,"title":"The Best Web Sources for Financial Report Research","slug":"the-best-web-sources-for-financial-report-research","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159959"}},{"articleId":159956,"title":"Is One International Financial Reporting Standard Likely?","slug":"is-one-international-financial-reporting-standard-likely","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159956"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282529,"slug":"reading-financial-reports-for-dummies","isbn":"9781119871361","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119871360/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119871360/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119871360-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119871360/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119871360/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/9781119871361-203x255.jpg","width":203,"height":255},"title":"Reading Financial Reports For Dummies","testBankPinActivationLink":"","bookOutOfPrint":true,"authorsInfo":"<p><b><b data-author-id=\"8974\">Lita Epstein</b>,</b> who earned her MBA from Emory University&#8217;s Goizueta Business School, enjoys helping people develop good financial, investing and tax-planning skills.<br /> While getting her MBA, Lita worked as a teaching assistant for the financial accounting department and ran the accounting lab. After completing her MBA, she managed finances for a small nonprofit organization and for the facilities management section of a large medical clinic.<br /> She designs and teaches online courses on topics such as investing for retirement, getting ready for tax time and finance and investing for women. She&#8217;s written over 20 books including <i>Reading Financial Reports For Dummies</i> and <i>Trading For Dummies.<br /> </i>Lita was the content director for a financial services Web site, MostChoice.com, and managed the Web site, Investing for Women. As a Congressional press secretary, Lita gained firsthand knowledge about how to work within and around the Federal bureaucracy, which gives her great insight into how government programs work. In the past, Lita has been a daily newspaper reporter, magazine editor, and fundraiser for the international activities of former President Jimmy Carter through The Carter Center.</p>","authors":[{"authorId":8974,"name":"Lita Epstein","slug":"lita-epstein","description":" <b>Lita Epstein,</b> who earned her MBA from Emory University&#8217;s Goizueta Business School, enjoys helping people develop good financial, investing and tax-planning skills.<br /> While getting her MBA, Lita worked as a teaching assistant for the financial accounting department and ran the accounting lab. After completing her MBA, she managed finances for a small nonprofit organization and for the facilities management section of a large medical clinic.<br /> She designs and teaches online courses on topics such as investing for retirement, getting ready for tax time and finance and investing for women. She&#8217;s written over 20 books including <i>Reading Financial Reports For Dummies</i> and <i>Trading For Dummies.<br /> </i>Lita was the content director for a financial services Web site, MostChoice.com, and managed the Web site, Investing for Women. As a Congressional press secretary, Lita gained firsthand knowledge about how to work within and around the Federal bureaucracy, which gives her great insight into how government programs work. In the past, Lita has been a daily newspaper reporter, magazine editor, and fundraiser for the international activities of former President Jimmy Carter through The Carter Center.","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8974"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119871361&quot;]}]\" id=\"du-slot-63221b1c735e2\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119871361&quot;]}]\" id=\"du-slot-63221b1c73fb4\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":159936,"title":"Key Parts in an Annual Report","slug":"key-parts-in-an-annual-report","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159936"}},{"articleId":159926,"title":"Key Securities and Exchange Commission Reports","slug":"key-securities-and-exchange-commission-reports","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159926"}},{"articleId":159954,"title":"Reading Financial Reports for Profitability Ratios","slug":"reading-financial-reports-for-profitability-ratios","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159954"}},{"articleId":159924,"title":"Reading Financial Reports for Liquidity Ratios","slug":"reading-financial-reports-for-liquidity-ratios","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159924"}},{"articleId":159925,"title":"Reading Financial Reports for Cash Flow Formulas","slug":"reading-financial-reports-for-cash-flow-formulas","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/159925"}}],"content":[{"title":"Key parts in an annual report","thumb":null,"image":null,"content":"<p>Annual reports can be daunting, and you may be relieved to know that you don’t actually need to scour every page of one. The following parts best serve to give you the big picture:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Auditor’s report:</b> Tells you whether the numbers are accurate and whether you should have any concerns about the future operation of the business</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Financial statements:</b> The balance sheet, the income statement, and the statement of cash flows; where you find the actual financial results for the year</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Notes to the financial statements:</b> Details about potential problems with the numbers or how the numbers were derived</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Management’s discussion and analysis:</b> The higher-ups’ breakdown of the financial results and other factors that impact the company’s operations</p>\n</li>\n</ul>\n<p>The rest is fluff.</p>\n"},{"title":"Key Securities and Exchange Commission Reports","thumb":null,"image":null,"content":"<p>Reports to the government are more extensive than the glossy reports sent to shareholders. Although many different types of forms must be filed with the Securities and Exchange Commission, you can get most of the juicy information from just a few:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>10-K:</b> Annual report that provides a comprehensive overview of the corporation’s business</p>\n</li>\n<li>\n<p class=\"first-para\"><b>10-Q:</b> Quarterly report that describes key financial information about the prior three months</p>\n</li>\n<li>\n<p class=\"first-para\"><b>8-K:</b> Shows any major events that could impact the financial position of the company</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Forms 3 to 5:</b> Reflect changes in ownership of stock by directors, officers, and major stockholders, giving you an idea of the view from the inside</p>\n</li>\n</ul>\n"},{"title":"Profitability ratios","thumb":null,"image":null,"content":"<p>You read financial reports to get a sense of a company&#8217;s financial position and how viable it is in the marketplace. You can test a company&#8217;s money-making prowess using the following important formulas.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Price/earnings ratio </i>compares the price of a stock to its earnings. A ratio of 10 means that for every $1 in company earnings per share, people are willing to pay $10 per share to buy the stock.</p>\n<p class=\"child-para\">Price/earnings ratio = Market value per share of stock divided by Earnings per share of stock</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Dividend payout ratio</i> shows the amount of a company&#8217;s earnings that are paid out to investors. Use it to determine the actual cash return you get by buying and holding a share of stock.</p>\n<p class=\"child-para\">Dividend payout ratio = Yearly dividend per share divided by Earnings per share</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on sales</i> tests how efficiently a company is running its operations by measuring the profit produced per dollar of sales.</p>\n<p class=\"child-para\">Return on sales = Net income before taxes divided by Sales</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on assets </i>shows you how well a company uses its assets. A high return on assets usually means the company is managing its assets well.</p>\n<p class=\"child-para\">Return on assets = Net income divided by Total assets</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Return on equity</i> measures how well a company earns money for its investors.</p>\n<p class=\"child-para\">Return on equity = Net income divided by Shareholders&#8217; equity</p>\n</li>\n<li>\n<p class=\"first-para\">The <i>gross margin</i> gives you a picture of how much revenue is left after all the direct costs of producing and selling the product have been subtracted.</p>\n<p class=\"child-para\">Gross margin = Gross profit divided by Net sales or revenues</p>\n</li>\n<li>\n<p class=\"first-para\">The <i>operating margin</i> looks at how well a company controls costs, factoring in any expenses not directly related to the production and sales of a particular product.</p>\n<p class=\"child-para\">Operating margin = Operating profit divided by Net sales or revenues</p>\n</li>\n</ul>\n"},{"title":"Liquidity ratios","thumb":null,"image":null,"content":"<p>If a company doesn’t have cash on hand to cover its day-to-day operations, it’s probably on shaky ground. Use the following formulas to find out whether a company has plenty of liquid (easily converted to cash) assets.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Current ratio</i> gives you a good idea of whether a company will be able to pay any bills due over the next 12 months with assets it has on hand.</p>\n<p class=\"child-para\">Current ratio = Current assets divided by Current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Quick ratio</i> or <i>acid test ratio</i> shows a company’s ability to pay its bills using only cash on hand or cash already due from accounts receivable. It doesn’t include money anticipated from the sale of inventory and the collection of the money from those sales.</p>\n<p class=\"child-para\">Quick ratio = Quick assets divided by Current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Interest coverage ratio</i> lets you know whether a company is bringing in enough money to pay interest on whatever outstanding debt it has.</p>\n<p class=\"child-para\">Interest coverage ratio = EBITDA divided by Interest expense</p>\n</li>\n</ul>\n"},{"title":"Cash flow formulas","thumb":null,"image":null,"content":"<p>You&#8217;re interested in a company, so you&#8217;re reading its financial reports. Part of the test of a viable operation is having enough cash to keep the company going. Use the following formulas to make sure a company has plenty of cash to keep operating.</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><i>Free cash flow</i> shows you how much money a company earns from its operations that can actually be put in a savings account for future use.</p>\n<p class=\"child-para\">Free cash flow = Cash provided by operating activities – Capital expenditures – Cash dividends</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Cash return on sales</i> looks specifically at how much cash is being generated by sales.</p>\n<p class=\"child-para\">Cash return on sales = Cash provided by operating activities divided by Net sales</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Current cash debt coverage ratio </i>lets you know whether a company has enough cash to meet its short-term needs.</p>\n<p class=\"child-para\">Current cash debt coverage ratio = Cash provided by operating activities divided by Average current liabilities</p>\n</li>\n<li>\n<p class=\"first-para\"><i>Cash flow coverage ratio</i> finds out whether a company has enough money to cover its bills <i>and</i> finance growth.</p>\n<p class=\"child-para\">Cash flow coverage ratio = Cash flows from operating activities divided by Cash requirements</p>\n</li>\n</ul>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Two years","lifeExpectancySetFrom":"2022-02-07T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207792},{"headers":{"creationTime":"2016-03-27T16:48:44+00:00","modifiedTime":"2022-01-31T19:37:09+00:00","timestamp":"2022-09-14T18:19:04+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Business Statistics For Dummies Cheat Sheet","strippedTitle":"business statistics for dummies cheat sheet","slug":"business-statistics-for-dummies-cheat-sheet","canonicalUrl":"","seo":{"metaDescription":"This Cheat Sheet is a handy resource for the basics of business statistics, including hypothesis testing, sampling distributions, and more.","noIndex":0,"noFollow":0},"content":"Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.","description":"Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.","blurb":"","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":162083,"title":"How Businesses Use Regression Analysis Statistics","slug":"how-businesses-use-regression-analysis-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162083"}},{"articleId":162074,"title":"Random Variables and Probability Distributions in Business Statistics","slug":"random-variables-and-probability-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162074"}},{"articleId":162073,"title":"Explore Hypothesis Testing in Business Statistics","slug":"explore-hypothesis-testing-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162073"}},{"articleId":162066,"title":"3 Ways to Describe Populations and Samples in Business Statistics","slug":"3-ways-to-describe-populations-and-samples-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162066"}},{"articleId":162062,"title":"Business Statistics: Assess Financial Risk with the VaR Methodology","slug":"business-statistics-assess-financial-risk-with-the-var-methodology","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162062"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282040,"slug":"business-statistics-for-dummies","isbn":"9781118630693","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1118630696-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1118630696/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/business-statistics-for-dummies-cover-9781118630693-203x255.jpg","width":203,"height":255},"title":"Business Statistics For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"9080\">Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","authors":[{"authorId":9080,"name":"Alan Anderson","slug":"alan-anderson","description":" <p><b>Alan Anderson</b>, PhD is a teacher of finance, economics, statistics, and math at Fordham and Fairfield universities as well as at Manhattanville and Purchase colleges. Outside of the academic environment he has many years of experience working as an economist, risk manager, and fixed income analyst. Alan received his PhD in economics from Fordham University, and an M.S. in financial engineering from Polytechnic University.</p>","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/9080"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-63221b185efa1\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781118630693&quot;]}]\" id=\"du-slot-63221b185f860\"></div></div>"},"articleType":{"articleType":"Cheat Sheet","articleList":[{"articleId":162066,"title":"3 Ways to Describe Populations and Samples in Business Statistics","slug":"3-ways-to-describe-populations-and-samples-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162066"}},{"articleId":162074,"title":"Random Variables and Probability Distributions in Business Statistics","slug":"random-variables-and-probability-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162074"}},{"articleId":162063,"title":"Understand Sampling Distributions in Business Statistics","slug":"understand-sampling-distributions-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162063"}},{"articleId":162073,"title":"Explore Hypothesis Testing in Business Statistics","slug":"explore-hypothesis-testing-in-business-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162073"}},{"articleId":162083,"title":"How Businesses Use Regression Analysis Statistics","slug":"how-businesses-use-regression-analysis-statistics","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/162083"}}],"content":[{"title":"3 ways to describe populations and samples","thumb":null,"image":null,"content":"<p>When you&#8217;re working with populations and samples (a subset of a population) in business statistics, you can use three common types of measures to describe the data set: central tendency, dispersion, and association.</p>\n<p class=\"Tip\">By convention, the statistical formulas used to describe population measures contain Greek letters, while the formulas used to describe sample measures contain Latin letters.</p>\n<h3>Measures of central tendency</h3>\n<p>In statistics, the mean, median, and mode are known as measures of <em>central tendency</em>; they are used to identify the center of a data set:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><strong>Mean:</strong> The value between the largest and smallest values of a data set, obtained by a prescribed method.</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Median: </strong>The value which divides a data set into two equal halves</p>\n</li>\n<li>\n<p class=\"first-para\"><strong>Mode:</strong> The most commonly observed value in a data set</p>\n</li>\n</ul>\n<p>Samples are randomly chosen from populations. If this process is carried out correctly, each sample should accurately reflect the characteristics of the population. So, a sample measure, such as the <em>mean,</em> should be a good estimate of the corresponding population measure. Consider the following examples of mean:</p>\n<p>Population mean:</p>\n<p>This formula simply tells you to add up all the elements in the population and divide by the size of the population.</p>\n<p>Sample mean:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393293.image0.jpg\" alt=\"image0.jpg\" width=\"106\" height=\"93\" /></p>\n<p>The process for computing this is exactly the same; you add up all the elements in the sample and divide by the size of the sample.</p>\n<p>In addition to measures of central tendency, two other key types of measures are measures of dispersion (spread) and measures of association.</p>\n<h3>Measures of dispersion</h3>\n<p><em>M</em><em>easures of dispersion</em> include variance/standard deviation and percentiles/quartiles/interquartile range. The variance and standard deviation are closely related to each other; the standard deviation always equals the <em>square root</em> of the variance.</p>\n<p>The formulas for the population and sample variance are:</p>\n<p>Population variance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393294.image1.jpg\" alt=\"image1.jpg\" width=\"159\" height=\"86\" /></p>\n<p>Sample variance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393295.image2.jpg\" alt=\"image2.jpg\" width=\"159\" height=\"86\" /></p>\n<p class=\"Remember\"><em>Percentiles</em> split up a data set into 100 equal parts each consisting of 1 percent of the values in the data set. <em>Quartiles</em> are a special type of percentiles; they split up the data into four equal parts. The <em>interquartile</em> range represents the middle 50 percent of the data; it&#8217;s calculated as the third quartile minus the first quartile.</p>\n<h3>Measures of association</h3>\n<p>Another type of measure, known as a <em>measure of </em><em>association</em>, refers to the <em>relationship </em>between two samples or two populations. Two examples of this are the <em>covariance</em> and the <em>correlation</em>:</p>\n<p>Population covariance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393296.image3.jpg\" alt=\"image3.jpg\" width=\"257\" height=\"86\" /></p>\n<p>Sample covariance:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393297.image4.jpg\" alt=\"image4.jpg\" width=\"257\" height=\"86\" /></p>\n<p>Population correlation:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393298.image5.jpg\" alt=\"image5.jpg\" width=\"127\" height=\"57\" /></p>\n<p>Sample correlation:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393299.image6.jpg\" alt=\"image6.jpg\" width=\"110\" height=\"61\" /></p>\n<p>The correlation is closely related to the covariance; it&#8217;s defined to ensure that its value is always between negative one and positive one.</p>\n"},{"title":"Random variables and probability distributions","thumb":null,"image":null,"content":"<p>Random variables and probability distributions are two of the most important concepts in statistics. A <i>random variable</i> assigns unique numerical values to the outcomes of a <i>random experiment; </i>this is a process that generates uncertain outcomes. A <i>probability distribution</i> assigns probabilities to each possible value of a random variable.</p>\n<p>The two basic types of probability distributions are discrete and continuous. A discrete probability distribution can only assume a <i>finite</i> number of different values.</p>\n<p>Examples of discrete distributions include:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Binomial</p>\n</li>\n<li>\n<p class=\"first-para\">Geometric</p>\n</li>\n<li>\n<p class=\"first-para\">Poisson</p>\n</li>\n</ul>\n<p>A continuous probability distribution can assume an <i>infinite</i> number of different values. Examples of continuous distributions include:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Uniform</p>\n</li>\n<li>\n<p class=\"first-para\">Normal</p>\n</li>\n<li>\n<p class=\"first-para\">Student&#8217;s t</p>\n</li>\n<li>\n<p class=\"first-para\">Chi-square</p>\n</li>\n<li>\n<p class=\"first-para\">F</p>\n</li>\n</ul>\n"},{"title":"Understand sampling distributions","thumb":null,"image":null,"content":"<p>In statistics, sampling distributions are the probability distributions of any given statistic based on a random sample, and are important because they provide a major simplification on the route to statistical inference. More specifically, they allow analytical considerations to be based on the sampling distribution of a statistic, rather than on the joint probability distribution of all the individual sample values.</p>\n<p class=\"Remember\">The value of a sample statistic such as the sample mean (X) is likely to be different for each sample that is drawn from a population. It can, therefore, be thought of as a <i>random variable</i>, whose properties can be described with a <i>probability distribution</i>. The probability distribution of a sample statistic is known as a <i>sampling distribution</i>.</p>\n<p>According to a key result in statistics known as the Central Limit Theorem, the sampling distribution of the sample mean is normal if one of two things is true:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">The underlying population is normal</p>\n</li>\n<li>\n<p class=\"first-para\">The sample size is at least 30</p>\n</li>\n</ul>\n<p>Two moments are needed to compute probabilities for the sample mean; the mean of the sampling distribution equals:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393317.image0.png\" alt=\"image0.png\" width=\"63\" height=\"32\" /></p>\n<p>The standard deviation of the sampling distribution (also known as the <i>standard error</i>) can take on one of two possible values:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393318.image1.png\" alt=\"image1.png\" width=\"76\" height=\"48\" /></p>\n<p>This is the appropriate choice for a &#8220;small&#8221; sample; for example, the sample size is less than or equal to 5 percent of the population size.</p>\n<p>If the sample is &#8220;large,&#8221; the standard error becomes:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393319.image2.png\" alt=\"image2.png\" width=\"123\" height=\"51\" /></p>\n<p>Probabilities may be computed for the sample mean directly from the standard normal table by applying the following formula:</p>\n<p><img loading=\"lazy\" src=\"https://www.dummies.com/wp-content/uploads/393320.image3.png\" alt=\"image3.png\" width=\"92\" height=\"51\" /></p>\n"},{"title":"Explore hypothesis testing in business statistics","thumb":null,"image":null,"content":"<p>In statistics, <i>h</i><i>ypothesis testing</i> refers to the process of choosing between competing hypotheses about a probability distribution, based on observed data from the distribution. It&#8217;s a core topic and a fundamental part of the language of statistics.</p>\n<p>Hypothesis testing is a six-step procedure:</p>\n<p>1.    Null hypothesis</p>\n<p>2.    Alternative hypothesis</p>\n<p>3.    Level of significance</p>\n<p>4.    Test statistic</p>\n<p>5.    Critical value(s)</p>\n<p>6.    Decision rule</p>\n<p>The <i>null hypothesis</i> is a statement that&#8217;s assumed to be true unless there&#8217;s strong contradictory evidence. The <i>alternative hypothesis</i> is a statement that will be accepted in place of the null hypothesis if it is rejected.</p>\n<p>The <i>level of significance</i> is chosen to control the probability of a &#8220;Type I&#8221; error; this is the error that results when the null hypothesis is erroneously rejected.</p>\n<p>The <i>test statistic </i>and<i> critical values</i> are used to determine if the null hypothesis should be rejected. The <i>decision rule</i> that is followed is that an &#8220;extreme&#8221; test statistic results in rejection of the null hypothesis. Here, an extreme test statistic is one that lies outside the bounds of the critical value or values.</p>\n<p class=\"Remember\">Hypotheses are often tested about the values of population measures such as the mean and the variance. They are also used to determine if a population follows a specified probability distribution. They also form a major part of regression analysis, where hypotheses are used to validate the results of an estimated regression equation.</p>\n"},{"title":"How businesses use regression analysis statistics","thumb":null,"image":null,"content":"<p>Regression analysis is a statistical tool used for the investigation of relationships between variables. Usually, the investigator seeks to ascertain the causal effect of one variable upon another — the effect of a price increase upon demand, for example, or the effect of changes in the money supply upon the inflation rate.</p>\n<p>Regression analysis is used to estimate the strength and the direction of the relationship between two linearly related variables: X and Y. X is the &#8220;independent&#8221; variable and Y is the &#8220;dependent&#8221; variable.</p>\n<p>The two basic types of regression analysis are:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\"><b>Simple regression analysis:</b> Used to estimate the relationship between a dependent variable and a single independent variable; for example, the relationship between crop yields and rainfall.</p>\n</li>\n<li>\n<p class=\"first-para\"><b>Multiple regression analysis:</b> Used to estimate the relationship between a dependent variable and two or more independent variables; for example, the relationship between the salaries of employees and their experience and education.</p>\n<p class=\"child-para\">Multiple regression analysis introduces several additional complexities but may produce more realistic results than simple regression analysis.</p>\n</li>\n</ul>\n<p>Regression analysis is based on several strong assumptions about the variables that are being estimated. Several key tests are used to ensure that the results are valid, including hypothesis tests. These tests are used to ensure that the regression results are not simply due to random chance but indicate an actual relationship between two or more variables.</p>\n<p>An estimated regression equation may be used for a wide variety of business applications, such as:</p>\n<ul class=\"level-one\">\n<li>\n<p class=\"first-para\">Measuring the impact on a corporation&#8217;s profits of an increase in profits</p>\n</li>\n<li>\n<p class=\"first-para\">Understanding how sensitive a corporation&#8217;s sales are to changes in advertising expenditures</p>\n</li>\n<li>\n<p class=\"first-para\">Seeing how a stock price is affected by changes in interest rates</p>\n</li>\n</ul>\n<p>Regression analysis may also be used for forecasting purposes; for example, a regression equation may be used to forecast the future demand for a company&#8217;s products.</p>\n<p>Due to the extreme complexity of regression analysis, it is often implemented through the use of specialized calculators or spreadsheet programs.</p>\n"}],"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":"Five years","lifeExpectancySetFrom":"2022-01-31T00:00:00+00:00","dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":207822},{"headers":{"creationTime":"2018-07-30T20:36:13+00:00","modifiedTime":"2019-06-20T18:42:50+00:00","timestamp":"2022-09-14T18:17:11+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"Important Terms in Game Theory","strippedTitle":"important terms in game theory","slug":"important-terms-game-theory","canonicalUrl":"","seo":{"metaDescription":"As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important a","noIndex":0,"noFollow":0},"content":"As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know.\r\n<ul>\r\n \t<li><b>Payoff matrix:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A matrix whose elements represent all the amounts won or lost by the row player.</span></li>\r\n \t<li><b>Payoff:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player.</span></li>\r\n \t<li><b>Saddle point:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points.</span></li>\r\n \t<li><b>Strictly determined game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game that has a saddle point.</span></li>\r\n \t<li><strong>Strategy:</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A move or moves chosen by a player.</span></li>\r\n \t<li><b>Optimal strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The strategy that most benefits a player.</span></li>\r\n \t<li><b>Value (expected value) of game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The amount representing the result when the best possible strategy is played by each player.</span></li>\r\n \t<li><b>Zero-sum game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game where what one player wins, the other loses; no money comes in from the outside or leaves.</span></li>\r\n \t<li><b>Fair game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game with a value of 0.</span></li>\r\n \t<li><b>Pure strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player always chooses the same row or column.</span></li>\r\n \t<li><b>Mixed strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player changes the choice of row or column with different plays or turns.</span></li>\r\n \t<li><b>Dominated strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.</span></li>\r\n</ul>","description":"As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know.\r\n<ul>\r\n \t<li><b>Payoff matrix:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A matrix whose elements represent all the amounts won or lost by the row player.</span></li>\r\n \t<li><b>Payoff:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player.</span></li>\r\n \t<li><b>Saddle point:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points.</span></li>\r\n \t<li><b>Strictly determined game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game that has a saddle point.</span></li>\r\n \t<li><strong>Strategy:</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A move or moves chosen by a player.</span></li>\r\n \t<li><b>Optimal strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The strategy that most benefits a player.</span></li>\r\n \t<li><b>Value (expected value) of game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> The amount representing the result when the best possible strategy is played by each player.</span></li>\r\n \t<li><b>Zero-sum game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game where what one player wins, the other loses; no money comes in from the outside or leaves.</span></li>\r\n \t<li><b>Fair game:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A game with a value of 0.</span></li>\r\n \t<li><b>Pure strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player always chooses the same row or column.</span></li>\r\n \t<li><b>Mixed strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A player changes the choice of row or column with different plays or turns.</span></li>\r\n \t<li><b>Dominated strategy:</b><span style=\"display: inline !important; float: none; background-color: transparent; color: #333333; cursor: text; font-family: Georgia,'Times New Roman','Bitstream Charter',Times,serif; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; line-height: 24px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\"> A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.</span></li>\r\n</ul>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221aa797539\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221aa797de9\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254831},{"headers":{"creationTime":"2018-07-30T20:31:43+00:00","modifiedTime":"2018-07-30T20:31:43+00:00","timestamp":"2022-09-14T18:16:42+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Create a Matrix from a Transition Diagram","strippedTitle":"how to create a matrix from a transition diagram","slug":"create-matrix-transition-diagram","canonicalUrl":"","seo":{"metaDescription":"When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The differ","noIndex":0,"noFollow":0},"content":"When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows.\r\n\r\nThe transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates.\r\n\r\n[caption id=\"attachment_254823\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254823\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1402.jpg\" alt=\"FNTMath-transition-diagram\" width=\"515\" height=\"363\" /> Eighty percent of the no-accident drivers probably won’t have an accident.[/caption]\r\n\r\nYou see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there.\r\n\r\nTo create a transition matrix representing the drivers, use the percentages to show going from one state to another.\r\n\r\n<img class=\"alignnone size-full wp-image-254726\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2401.jpg\" alt=\"FNTMATH_2401\" width=\"264\" height=\"100\" />\r\n\r\nWhat is the long-term expectation for these drivers? First, let the transition matrix be D.\r\n\r\n<img class=\"alignnone size-full wp-image-254727\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2402.jpg\" alt=\"FNTMATH_2402\" width=\"155\" height=\"68\" />\r\n\r\nThen, some of the powers of D are\r\n\r\n<img class=\"alignnone size-full wp-image-254728\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2403.jpg\" alt=\"FNTMATH_2403\" width=\"253\" height=\"297\" />\r\n\r\nAt the end of ten years, using the drivers in the initial study, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254729\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2404.jpg\" alt=\"FNTMATH_2404\" width=\"267\" height=\"100\" />\r\n\r\nWhat this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.","description":"When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows.\r\n\r\nThe transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates.\r\n\r\n[caption id=\"attachment_254823\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254823\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1402.jpg\" alt=\"FNTMath-transition-diagram\" width=\"515\" height=\"363\" /> Eighty percent of the no-accident drivers probably won’t have an accident.[/caption]\r\n\r\nYou see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there.\r\n\r\nTo create a transition matrix representing the drivers, use the percentages to show going from one state to another.\r\n\r\n<img class=\"alignnone size-full wp-image-254726\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2401.jpg\" alt=\"FNTMATH_2401\" width=\"264\" height=\"100\" />\r\n\r\nWhat is the long-term expectation for these drivers? First, let the transition matrix be D.\r\n\r\n<img class=\"alignnone size-full wp-image-254727\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2402.jpg\" alt=\"FNTMATH_2402\" width=\"155\" height=\"68\" />\r\n\r\nThen, some of the powers of D are\r\n\r\n<img class=\"alignnone size-full wp-image-254728\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2403.jpg\" alt=\"FNTMATH_2403\" width=\"253\" height=\"297\" />\r\n\r\nAt the end of ten years, using the drivers in the initial study, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254729\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2404.jpg\" alt=\"FNTMATH_2404\" width=\"267\" height=\"100\" />\r\n\r\nWhat this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221a8a2f9d2\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221a8a302dd\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254827},{"headers":{"creationTime":"2018-07-30T20:26:00+00:00","modifiedTime":"2018-07-30T20:26:00+00:00","timestamp":"2022-09-14T18:16:42+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Use Transition Matrices","strippedTitle":"how to use transition matrices","slug":"use-transition-matrices","canonicalUrl":"","seo":{"metaDescription":"If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A transi","noIndex":0,"noFollow":0},"content":"If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A <em>transition matrix</em> consists of a square matrix that gives the probabilities of different states going from one to another.\r\n\r\nWith a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications.\r\n\r\nConsider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run?\r\n\r\n<img class=\"alignnone size-full wp-image-254717\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2301.jpg\" alt=\"FNTMATH_2301\" width=\"363\" height=\"103\" />\r\n\r\nOne way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases.\r\n\r\n[caption id=\"attachment_254822\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254822\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1401.jpg\" alt=\"FNTMath-transition-matrices\" width=\"515\" height=\"320\" /> Which kind of cereal will the consumer buy?[/caption]\r\n\r\nIf you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time.\r\n\r\nThe tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values.\r\n\r\nTo perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C.\r\n\r\n<img class=\"alignnone size-full wp-image-254718\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2302.jpg\" alt=\"FNTMATH_2302\" width=\"155\" height=\"68\" />\r\n\r\nNext, use matrix multiplication to find <em>C</em>². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled <em>c</em>11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products.\r\n<p class=\"article-tips remember\">In a matrix A, the element in the <em>n</em>th row, <em>k</em>th column is labeled <em>ank</em>.</p>\r\nThe element in the first row and second column of the product, <em>c</em>12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements.\r\n\r\n<img class=\"alignnone size-full wp-image-254719\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2303.jpg\" alt=\"FNTMATH_2303\" width=\"316\" height=\"68\" />\r\n\r\nSo, you take the first row of the left matrix times the first column of the second matrix to get\r\n\r\n<img class=\"alignnone size-full wp-image-254720\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2304.jpg\" alt=\"FNTMATH_2304\" width=\"207\" height=\"40\" />\r\n\r\nYes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases.\r\n\r\nPerforming the matrix multiplication, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254721\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2305.jpg\" alt=\"FNTMATH_2305\" width=\"191\" height=\"68\" />\r\n\r\nContinuing this multiplication process, by the time <em>C</em>6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges.\r\n\r\n<img class=\"alignnone size-full wp-image-254722\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2306.jpg\" alt=\"FNTMATH_2306\" width=\"249\" height=\"68\" />\r\n\r\nNotice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes\r\n\r\n<img class=\"alignnone size-full wp-image-254723\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2307.jpg\" alt=\"FNTMATH_2307\" width=\"184\" height=\"68\" />\r\n\r\nThe matrix shows you the pattern or trend.\r\n\r\n<img class=\"alignnone size-full wp-image-254724\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2308.jpg\" alt=\"FNTMATH_2308\" width=\"239\" height=\"100\" />\r\n\r\nNo matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:\r\n\r\n<img class=\"alignnone size-full wp-image-254725\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2309.jpg\" alt=\"FNTMATH_2309\" width=\"153\" height=\"65\" />","description":"If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A <em>transition matrix</em> consists of a square matrix that gives the probabilities of different states going from one to another.\r\n\r\nWith a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications.\r\n\r\nConsider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run?\r\n\r\n<img class=\"alignnone size-full wp-image-254717\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2301.jpg\" alt=\"FNTMATH_2301\" width=\"363\" height=\"103\" />\r\n\r\nOne way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases.\r\n\r\n[caption id=\"attachment_254822\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254822\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1401.jpg\" alt=\"FNTMath-transition-matrices\" width=\"515\" height=\"320\" /> Which kind of cereal will the consumer buy?[/caption]\r\n\r\nIf you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time.\r\n\r\nThe tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values.\r\n\r\nTo perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C.\r\n\r\n<img class=\"alignnone size-full wp-image-254718\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2302.jpg\" alt=\"FNTMATH_2302\" width=\"155\" height=\"68\" />\r\n\r\nNext, use matrix multiplication to find <em>C</em>². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled <em>c</em>11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products.\r\n<p class=\"article-tips remember\">In a matrix A, the element in the <em>n</em>th row, <em>k</em>th column is labeled <em>ank</em>.</p>\r\nThe element in the first row and second column of the product, <em>c</em>12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements.\r\n\r\n<img class=\"alignnone size-full wp-image-254719\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2303.jpg\" alt=\"FNTMATH_2303\" width=\"316\" height=\"68\" />\r\n\r\nSo, you take the first row of the left matrix times the first column of the second matrix to get\r\n\r\n<img class=\"alignnone size-full wp-image-254720\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2304.jpg\" alt=\"FNTMATH_2304\" width=\"207\" height=\"40\" />\r\n\r\nYes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases.\r\n\r\nPerforming the matrix multiplication, you have\r\n\r\n<img class=\"alignnone size-full wp-image-254721\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2305.jpg\" alt=\"FNTMATH_2305\" width=\"191\" height=\"68\" />\r\n\r\nContinuing this multiplication process, by the time <em>C</em>6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges.\r\n\r\n<img class=\"alignnone size-full wp-image-254722\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2306.jpg\" alt=\"FNTMATH_2306\" width=\"249\" height=\"68\" />\r\n\r\nNotice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes\r\n\r\n<img class=\"alignnone size-full wp-image-254723\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2307.jpg\" alt=\"FNTMATH_2307\" width=\"184\" height=\"68\" />\r\n\r\nThe matrix shows you the pattern or trend.\r\n\r\n<img class=\"alignnone size-full wp-image-254724\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2308.jpg\" alt=\"FNTMATH_2308\" width=\"239\" height=\"100\" />\r\n\r\nNo matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:\r\n\r\n<img class=\"alignnone size-full wp-image-254725\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2309.jpg\" alt=\"FNTMATH_2309\" width=\"153\" height=\"65\" />","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"_links":{"self":"https://dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221a8a271b1\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{&quot;key&quot;:&quot;cat&quot;,&quot;values&quot;:[&quot;business-careers-money&quot;,&quot;business&quot;,&quot;accounting&quot;,&quot;calculation-analysis&quot;]},{&quot;key&quot;:&quot;isbn&quot;,&quot;values&quot;:[&quot;9781119476368&quot;]}]\" id=\"du-slot-63221a8a27a45\"></div></div>"},"articleType":{"articleType":"Articles","articleList":null,"content":null,"videoInfo":{"videoId":null,"name":null,"accountId":null,"playerId":null,"thumbnailUrl":null,"description":null,"uploadDate":null}},"sponsorship":{"sponsorshipPage":false,"backgroundImage":{"src":null,"width":0,"height":0},"brandingLine":"","brandingLink":"","brandingLogo":{"src":null,"width":0,"height":0},"sponsorAd":"","sponsorEbookTitle":"","sponsorEbookLink":"","sponsorEbookImage":{"src":null,"width":0,"height":0}},"primaryLearningPath":"Advance","lifeExpectancy":null,"lifeExpectancySetFrom":null,"dummiesForKids":"no","sponsoredContent":"no","adInfo":"","adPairKey":[]},"status":"publish","visibility":"public","articleId":254821},{"headers":{"creationTime":"2018-07-30T20:13:00+00:00","modifiedTime":"2018-07-30T20:13:00+00:00","timestamp":"2022-09-14T18:16:42+00:00"},"data":{"breadcrumbs":[{"name":"Business, Careers, & Money","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34224"},"slug":"business-careers-money","categoryId":34224},{"name":"Business","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34225"},"slug":"business","categoryId":34225},{"name":"Accounting","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34226"},"slug":"accounting","categoryId":34226},{"name":"Calculation & Analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"},"slug":"calculation-analysis","categoryId":34229}],"title":"How to Analyze Arguments with Euler Diagrams","strippedTitle":"how to analyze arguments with euler diagrams","slug":"analyze-arguments-euler-diagrams","canonicalUrl":"","seo":{"metaDescription":"On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see w","noIndex":0,"noFollow":0},"content":"On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid.\r\n<p class=\"article-tips remember\">An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false.</p>\r\nThe following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas.\r\n\r\nThese arguments usually have the following format with the premises listed first and the conclusion under a horizontal line:\r\n\r\n<img class=\"alignnone size-full wp-image-254714\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2201.jpg\" alt=\"FNTMATH_2201\" width=\"199\" height=\"79\" />\r\n\r\nUsing an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach.\r\n\r\n[caption id=\"attachment_254815\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254815\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1301.jpg\" alt=\"1119476364-fg1301\" width=\"515\" height=\"543\" /> Poor Hank has fleas.[/caption]\r\n\r\nThe argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is <em>valid</em>. If the two premises are true, then the conclusion must be true.\r\n\r\nNow consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints.\r\n\r\n<img class=\"alignnone size-full wp-image-254715\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2202.jpg\" alt=\"FNTMATH_2202\" width=\"172\" height=\"81\" />\r\n\r\nWhen analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here.\r\n\r\n[caption id=\"attachment_254816\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254816\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1302.jpg\" alt=\"1119476364-fg1302\" width=\"515\" height=\"543\" /> Two types of polygons.[/caption]\r\n\r\nTwo circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides.\r\n\r\nThe argument is <em>invalid</em>. Rectangles are not triangles—not even sometimes.\r\n\r\nArguments can have more than two premises. For example:\r\n\r\n<img class=\"alignnone size-full wp-image-254716\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2203.jpg\" alt=\"FNTMATH_2203\" width=\"307\" height=\"112\" />\r\n\r\nOne Euler diagram that can represent this situation has three intersecting circles, as shown here.\r\n\r\n[caption id=\"attachment_254817\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254817\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1303.jpg\" alt=\"1119476364-fg1303\" width=\"515\" height=\"558\" /> President Abraham Lincoln and other Illinois lawyers.[/caption]\r\n\r\nAs you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is <em>invalid</em>. To be valid, it must always be true.","description":"On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid.\r\n<p class=\"article-tips remember\">An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false.</p>\r\nThe following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas.\r\n\r\nThese arguments usually have the following format with the premises listed first and the conclusion under a horizontal line:\r\n\r\n<img class=\"alignnone size-full wp-image-254714\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2201.jpg\" alt=\"FNTMATH_2201\" width=\"199\" height=\"79\" />\r\n\r\nUsing an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach.\r\n\r\n[caption id=\"attachment_254815\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254815\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1301.jpg\" alt=\"1119476364-fg1301\" width=\"515\" height=\"543\" /> Poor Hank has fleas.[/caption]\r\n\r\nThe argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is <em>valid</em>. If the two premises are true, then the conclusion must be true.\r\n\r\nNow consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints.\r\n\r\n<img class=\"alignnone size-full wp-image-254715\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2202.jpg\" alt=\"FNTMATH_2202\" width=\"172\" height=\"81\" />\r\n\r\nWhen analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here.\r\n\r\n[caption id=\"attachment_254816\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254816\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1302.jpg\" alt=\"1119476364-fg1302\" width=\"515\" height=\"543\" /> Two types of polygons.[/caption]\r\n\r\nTwo circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides.\r\n\r\nThe argument is <em>invalid</em>. Rectangles are not triangles—not even sometimes.\r\n\r\nArguments can have more than two premises. For example:\r\n\r\n<img class=\"alignnone size-full wp-image-254716\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2203.jpg\" alt=\"FNTMATH_2203\" width=\"307\" height=\"112\" />\r\n\r\nOne Euler diagram that can represent this situation has three intersecting circles, as shown here.\r\n\r\n[caption id=\"attachment_254817\" align=\"alignnone\" width=\"515\"]<img class=\"size-full wp-image-254817\" src=\"https://www.dummies.com/wp-content/uploads/1119476364-fg1303.jpg\" alt=\"1119476364-fg1303\" width=\"515\" height=\"558\" /> President Abraham Lincoln and other Illinois lawyers.[/caption]\r\n\r\nAs you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is <em>invalid</em>. To be valid, it must always be true.","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254811,"title":"How to Analyze Compound Statements","slug":"analyze-compound-statements","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254811"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. 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Not every topic in a discussion can be turn","noIndex":0,"noFollow":0},"content":"If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible.\r\n\r\nConsider the compound statement\r\n\r\n<img class=\"alignnone size-full wp-image-254706\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2101.jpg\" alt=\"FNTMATH_2101\" width=\"84\" height=\"25\" />\r\n\r\nWhen constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column\r\n\r\n<img class=\"alignnone size-full wp-image-254707\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2102.jpg\" alt=\"FNTMATH_2102\" width=\"45\" height=\"17\" />\r\n\r\nBefore you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure.\r\n<ol>\r\n \t<li><strong>Start with a basic <em>p</em> and<em> q</em> and then add ~ <em>q</em>.</strong>\r\n<img class=\"alignnone size-full wp-image-254708\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2103.jpg\" alt=\"FNTMATH_2103\" width=\"132\" height=\"100\" /></li>\r\n \t<li>When adding the\r\n<img class=\"alignnone size-full wp-image-254709\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2104.jpg\" alt=\"FNTMATH_2104\" width=\"45\" height=\"17\" />\r\ncolumn, perform the disjunction on the first and third columns.\r\nRemember, with disjunctions, the statement is false only when both component statements are false.\r\n<img class=\"alignnone size-full wp-image-254710\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2105.jpg\" alt=\"FNTMATH_2105\" width=\"131\" height=\"120\" /></li>\r\n \t<li><strong>Add the ~ p column.</strong>\r\n<img class=\"alignnone size-full wp-image-254711\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2106.jpg\" alt=\"FNTMATH_2106\" width=\"161\" height=\"100\" /></li>\r\n \t<li>Add the\r\n<img class=\"alignnone size-full wp-image-254712\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2107.jpg\" alt=\"FNTMATH_2107\" width=\"85\" height=\"25\" />\r\ncolumn, which shows the conjunction of the fourth and fifth columns.\r\n<img class=\"alignnone size-full wp-image-254713\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2108.jpg\" alt=\"FNTMATH_2108\" width=\"255\" height=\"124\" /></li>\r\n</ol>\r\n<p class=\"article-tips remember\">The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.</p>","description":"If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible.\r\n\r\nConsider the compound statement\r\n\r\n<img class=\"alignnone size-full wp-image-254706\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2101.jpg\" alt=\"FNTMATH_2101\" width=\"84\" height=\"25\" />\r\n\r\nWhen constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column\r\n\r\n<img class=\"alignnone size-full wp-image-254707\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2102.jpg\" alt=\"FNTMATH_2102\" width=\"45\" height=\"17\" />\r\n\r\nBefore you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure.\r\n<ol>\r\n \t<li><strong>Start with a basic <em>p</em> and<em> q</em> and then add ~ <em>q</em>.</strong>\r\n<img class=\"alignnone size-full wp-image-254708\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2103.jpg\" alt=\"FNTMATH_2103\" width=\"132\" height=\"100\" /></li>\r\n \t<li>When adding the\r\n<img class=\"alignnone size-full wp-image-254709\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2104.jpg\" alt=\"FNTMATH_2104\" width=\"45\" height=\"17\" />\r\ncolumn, perform the disjunction on the first and third columns.\r\nRemember, with disjunctions, the statement is false only when both component statements are false.\r\n<img class=\"alignnone size-full wp-image-254710\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2105.jpg\" alt=\"FNTMATH_2105\" width=\"131\" height=\"120\" /></li>\r\n \t<li><strong>Add the ~ p column.</strong>\r\n<img class=\"alignnone size-full wp-image-254711\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2106.jpg\" alt=\"FNTMATH_2106\" width=\"161\" height=\"100\" /></li>\r\n \t<li>Add the\r\n<img class=\"alignnone size-full wp-image-254712\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2107.jpg\" alt=\"FNTMATH_2107\" width=\"85\" height=\"25\" />\r\ncolumn, which shows the conjunction of the fourth and fifth columns.\r\n<img class=\"alignnone size-full wp-image-254713\" src=\"https://www.dummies.com/wp-content/uploads/FNTMATH_2108.jpg\" alt=\"FNTMATH_2108\" width=\"255\" height=\"124\" /></li>\r\n</ol>\r\n<p class=\"article-tips remember\">The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.</p>","blurb":"","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. ","hasArticle":false,"_links":{"self":"https://dummies-api.dummies.com/v2/authors/8985"}}],"primaryCategoryTaxonomy":{"categoryId":34229,"title":"Calculation & Analysis","slug":"calculation-analysis","_links":{"self":"https://dummies-api.dummies.com/v2/categories/34229"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[],"relatedArticles":{"fromBook":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}],"fromCategory":[{"articleId":254831,"title":"Important Terms in Game Theory","slug":"important-terms-game-theory","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254831"}},{"articleId":254827,"title":"How to Create a Matrix from a Transition Diagram","slug":"create-matrix-transition-diagram","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254827"}},{"articleId":254821,"title":"How to Use Transition Matrices","slug":"use-transition-matrices","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254821"}},{"articleId":254814,"title":"How to Analyze Arguments with Euler Diagrams","slug":"analyze-arguments-euler-diagrams","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254814"}},{"articleId":254808,"title":"How to Calculate Monthly Payments for a Sinking Fund","slug":"calculate-monthly-payments-sinking-fund","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"_links":{"self":"https://dummies-api.dummies.com/v2/articles/254808"}}]},"hasRelatedBookFromSearch":false,"relatedBook":{"bookId":282200,"slug":"finite-math-for-dummies","isbn":"9781119476368","categoryList":["business-careers-money","business","accounting","calculation-analysis"],"amazon":{"default":"https://www.amazon.com/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"https://www.amazon.ca/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119476364-item.html&cjsku=978111945484","gb":"https://www.amazon.co.uk/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"https://www.amazon.de/gp/product/1119476364/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"https://www.dummies.com/wp-content/uploads/finite-math-for-dummies-cover-9781119476368-203x255.jpg","width":203,"height":255},"title":"Finite Math For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8985\">Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. </p>","authors":[{"authorId":8985,"name":"Mary Jane Sterling","slug":"mary-jane-sterling","description":" <p><b>Mary Jane Sterling</b> is the author of <i>Algebra I For Dummies, Algebra Workbook For Dummies,</i> and many other <i>For Dummies</i> books. She taught at Bradley University in Peoria, Illinois for more than 30 years, teaching algebra, business calculus, geometry, and finite mathematics. 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Whether you're trying to figure out the difference between covariance and correlation or breaking out a regression equation, our articles have the info you need.

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216 results
Calculation & Analysis Business Statistics: Test the Estimated Regression Equation

Article / Updated 10-06-2022

After you estimate the population regression line, you can check whether the regression equation makes sense by using the coefficient of determination, also known as R2 (R squared). This is used as a measure of how well the regression equation actually describes the relationship between the dependent variable (Y) and the independent variable (X). It may be the case that there is no real relationship between the dependent and independent variables; simple regression generates results even if this is the case. It is, therefore, important to subject the regression results to some key tests that enable you to determine if the results are reliable. The coefficient of determination, R2, is a statistical measure that shows the proportion of variation explained by the estimated regression line. Variation refers to the sum of the squared differences between the values of Y and the mean value of Y, expressed mathematically as R2 always takes on a value between 0 and 1. The closer R2 is to 1, the better the estimated regression equation fits or explains the relationship between X and Y. The expression is also known as the total sum of squares (TSS). This sum can be divided into the following two categories: Explained sum of squares (ESS): Also known as the explained variation, the ESS is the portion of total variation that measures how well the regression equation explains the relationship between X and Y. You compute the ESS with the formula Residual sum of squares (RSS): This expression is also known as unexplained variation and is the portion of total variation that measures discrepancies (errors) between the actual values of Y and those estimated by the regression equation. You compute the RSS with the formula The smaller the value of RSS relative to ESS, the better the regression line fits or explains the relationship between the dependent and independent variable. Total sum of squares (TSS): The sum of RSS and ESS equals TSS. R2 is the ratio of explained sum of squares (ESS) to total sum of squares (TSS): You can also use this formula: Based on the definition of R2, its value can never be negative. Also, R2 can't be greater than 1, so With simple regression analysis, R2 equals the square of the correlation between X and Y. The coefficient of determination is used as a measure of how well a regression line explains the relationship between a dependent variable (Y) and an independent variable (X). The closer the coefficient of determination is to 1, the more closely the regression line fits the sample data. The coefficient of determination is computed from the sums of squares. These calculations are summarized in the following table. To compute ESS, you subtract the mean value of Y from each of the estimated values of Y; each term is squared and then added together: To compute RSS, you subtract the estimated value of Y from each of the actual values of Y; each term is squared and then added together: To compute TSS, you subtract the mean value of Y from each of the actual values of Y; each term is squared and then added together: Alternatively, you can simply add ESS and RSS to obtain TSS: TSS = ESS + RSS = 0.54 + 0.14 = 0.68 The coefficient of determination (R2) is the ratio of ESS to TSS: This shows that 79.41 percent of the variation in Y is explained by variation in X. Because the coefficient of determination can't exceed 100 percent, a value of 79.41 indicates that the regression line closely matches the actual sample data.

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Calculation & Analysis How to Convert a Sampling Distribution Using the Central Limit Theorem

Article / Updated 09-22-2022

You can use the Central Limit Theorem to convert a sampling distribution to a standard normal random variable. Based on the Central Limit Theorem, if you draw samples from a population that is greater than or equal to 30, then the sample mean is a normally distributed random variable. To determine probabilities for the sample mean the standard normal tables requires you to convert to a standard normal random variable. The standard normal distribution is the special case where the mean equals 0, and the standard deviation equals 1. For any normally distributed random variable X with a mean and a standard deviation you find the corresponding standard normal random variable (Z) with the following equation: For the sampling distribution of the corresponding equation is As an example, say that there are 10,000 stocks trading each day on a regional stock exchange. It's known from historical experience that the returns to these stocks have a mean value of 10 percent per year, and a standard deviation of 20 percent per year. An investor chooses to buy a random selection of 100 of these stocks for his portfolio. What's the probability that the mean rate of return among these 100 stocks is greater than 8 percent? The investor's portfolio can be thought of as a sample of stocks chosen from the population of stocks trading on the regional exchange. The first step to finding this probability is to compute the moments of the sampling distribution. Compute the mean: The mean of the sampling distribution equals the population mean. Determine the standard error: This calculation is a little trickier because the standard error depends on the size of the sample relative to the size of the population. In this case, the sample size (n) is 100, while the population size (N) is 10,000. So you first have to compute the sample size relative to the population size, like so: Because 1 percent is less than 5 percent, you don't use the finite population correction factor to compute the standard error. Note that in this case, the value of the finite population correction factor is: Because this value is so close to 1, using the finite population correction factor in this case would have little or no impact on the resulting probabilities. And because the finite population correction factor isn't needed in this case, the standard error is computed as follows: To determine the probability that the sample mean is greater than 8 percent, you must now convert the sample mean into a standard normal random variable using the following equation: To compute the probability that the sample mean is greater than 8 percent, you apply the previous formula as follows: Because these values are substituted into the previous expression as follows: You can calculate this probability by using the properties of the standard normal distribution along with a standard normal table such as this one. Standard Normal Table — Negative Values Z 0.00 0.01 0.02 0.03 –1.3 0.0968 0.0951 0.0934 0.0918 –1.2 0.1151 0.1131 0.1112 0.1093 –1.1 0.1357 0.1335 0.1314 0.1292 –1.0 0.1587 0.1562 0.1539 0.1515 The table shows the probability that a standard normal random variable (designated Z) is less than or equal to a specific value. For example, you can write the probability that (one standard deviation below the mean) as You find the probability from the table with these steps: Locate the first digit before and after the decimal point (–1.0) in the first (Z) column. Find the second digit after the decimal point (0.00) in the second (0.00) column. See where the row and column intersect to find the probability: Because you're actually looking for the probability that Z is greater than or equal to –1, one more step is required. Due to the symmetry of the standard normal distribution, the probability that Z is greater than or equal to a negative value equals one minus the probability that Z is less than or equal to the same negative value. For example, This is because are complementary events. This means that Z must either be greater than or equal to –2 or less than or equal to –2. Therefore, This is true because the occurrence of one of these events is certain, and the probability of a certain event is 1. After algebraically rewriting this equation, you end up with the following result: For the portfolio example, The result shows that there's an 84.13 percent chance that the investor's portfolio will have a mean return greater than 8 percent.

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Calculation & Analysis Finite Math For Dummies Cheat Sheet

Cheat Sheet / Updated 03-08-2022

When performing the many types of computations found in Finite Math topics, it’s helpful to have some numbers, notations, distributions, and listings right at hand.

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Calculation & Analysis Reading Financial Reports For Dummies Cheat Sheet

Cheat Sheet / Updated 02-16-2022

If you're looking at a business with an interest in investing in it, you need to read its financial reports. Of course, when it comes to the annual report, you don't need to read everything, just the key parts. Combining the annual report with some of the financial reports a corporation files with the Securities and Exchange Commission (SEC) can help you figure profitability and liquidity ratios and get a better sense of cash flow. Keep this handy Cheat Sheet nearby for a quick reference to reading financial reports, including SEC reports, profitability ratios, liquidity ratios, and cash flow formulas.

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Calculation & Analysis Business Statistics For Dummies Cheat Sheet

Cheat Sheet / Updated 01-31-2022

Statistics make it possible to analyze real-world business problems with actual data so that you can determine if a marketing strategy is really working, how much a company should charge for its products, or any of a million other practical questions. The science of statistics uses regression analysis, hypothesis testing, sampling distributions, and more to ensure accurate data analysis.

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Calculation & Analysis Important Terms in Game Theory

Article / Updated 06-20-2019

As with many areas and topics in finite mathematics, there is a very special and specific vocabulary that goes along with game theory. Here are some important and useful terms that you should know. Payoff matrix: A matrix whose elements represent all the amounts won or lost by the row player. Payoff: An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player. Saddle point: The element in a payoff matrix that is the smallest in a particular row while, at the same time, the largest in its column. Not all matrices have saddle points. Strictly determined game: A game that has a saddle point. Strategy: A move or moves chosen by a player. Optimal strategy: The strategy that most benefits a player. Value (expected value) of game: The amount representing the result when the best possible strategy is played by each player. Zero-sum game: A game where what one player wins, the other loses; no money comes in from the outside or leaves. Fair game: A game with a value of 0. Pure strategy: A player always chooses the same row or column. Mixed strategy: A player changes the choice of row or column with different plays or turns. Dominated strategy: A strategy that is never considered because another play is always better. For the row player, a row is dominated by another row if all the corresponding elements are all larger. For the column player, a column is dominated by another column if all the corresponding elements are all smaller.

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Calculation & Analysis How to Create a Matrix from a Transition Diagram

Article / Updated 07-30-2018

When you encounter a matrix problem in finite math, a nice way to illustrate the transition from one state to another is to use a transition diagram. The different states are represented by circles, and the probability of going from one state to another is shown by using curves with arrows. The transition diagram in the following figure shows how an insurance company classifies its drivers: no accidents, one accident, or two or more accidents. This information could help the company determine the insurance premium rates. You see that 80% of the drivers who haven’t had an accident aren’t expected to have an accident the next year. Fifteen percent of those drivers have one accident, and 5% have two or more accidents. Seventy percent of those who have had one accident aren’t expected to have an accident the next year but have to stay in the one-accident classification. And those in the two-or-more accident class have to stay there. To create a transition matrix representing the drivers, use the percentages to show going from one state to another. What is the long-term expectation for these drivers? First, let the transition matrix be D. Then, some of the powers of D are At the end of ten years, using the drivers in the initial study, you have What this tells the insurance company is that, in ten years, about 11% of the original no-accident drivers will still not have had an accident. Only 3% of the one-accident drivers will still have had only that one accident. This situation doesn’t allow for the drivers to move back or earn forgiveness; a one-accident driver can’t be a no-accident driver using this model. Of course, different insurance agencies have different policies, putting drivers in better standing after a set number of accident-free years. And new policyholders are added to make this picture rosier. This just shows the pattern for a particular set of drivers after a certain number of years.

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Calculation & Analysis How to Use Transition Matrices

Article / Updated 07-30-2018

If your finite math instructor asks you to predict the likelihood of an action repeating over time, you may need to use a transition matrix to do this. A transition matrix consists of a square matrix that gives the probabilities of different states going from one to another. With a transition matrix, you can perform matrix multiplication and determine trends, if there are any, and make predications. Consider the table showing the purchasing patterns involving different cereals. You see all the percentages showing the probability of going from one state to another, but which of the cereals does the consumer actually end up buying most frequently in the long run? One way to look at continued purchasing is to create a tree diagram. In the following figure, you see two consecutive “rounds” of purchases. If you want the probability that the consumer purchases Kicks first, tries it again or something else, and then purchases Kicks the next time, add up the , , and branches: , or 38% of the time. If you want the probability that the consumer purchases Cheery A’s first, tries something else or repeats Cheery A’s, and then tries Corn Flecks, add up the , , and branches. This comes out to , or almost 26% of the time. The tree is helpful in that it shows you what the choices are and how the percentages work in determining patterns, but there’s a much easier and neater way to compute these values. To perform computations and study this further, create a transition matrix, referring back to the chart showing purchases and using the decimal values of the percentages. Name it matrix C. Next, use matrix multiplication to find C². As a quick hint, when multiplying matrices, you find the element in the first row, first column of the product, labeled c11, when you multiply the elements in the first row of the first matrix times the corresponding elements in the first column of the second matrix and then add up the products. In a matrix A, the element in the nth row, kth column is labeled ank. The element in the first row and second column of the product, c12, uses the elements in the first row of the first matrix and second column of the second matrix, and so on for the rest of the elements. So, you take the first row of the left matrix times the first column of the second matrix to get Yes. This is the same computation as was done using the tree to find the probability that a consumer starting with Kicks would return to it in two more purchases. Performing the matrix multiplication, you have Continuing this multiplication process, by the time C6 appears (the chances of buying a particular cereal at the fifth purchase time after the initial purchase), a pattern emerges. Notice that the numbers in each column round to the same three decimal places. This is going to become even clearer, using higher powers of C, until some nth matrix power becomes The matrix shows you the pattern or trend. No matter which cereal the consumer bought first, in the long run there’s a 35.3% chance that she’ll purchase Kicks, a 38.4% chance that she’ll purchase Cheery A’s, and a 26.3% chance that she’ll purchase Corn Flecks. This transition matrix has reached an equilibrium, where it won’t change with more repeated multiplication. You can write this situation with a single-line matrix:

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Calculation & Analysis How to Analyze Arguments with Euler Diagrams

Article / Updated 07-30-2018

On a finite math exam, you may be asked to analyze an argument with a visual approach using an Euler diagram. This pictorial technique is used to check to see whether an argument is valid. An argument can be classified as either valid or invalid. A valid argument occurs in situations where if the premises are true, then the conclusion must also be true. And an argument can be valid even if the conclusion is false. The following argument has two premises: (1) “All dogs have fleas.” (2) “Hank is a dog.” The conclusion is that, therefore, Hank has fleas. These arguments usually have the following format with the premises listed first and the conclusion under a horizontal line: Using an Euler diagram to analyze this argument, draw a circle to contain all objects that have fleas. Inside the circle, put another circle to contain all dogs. And inside the circle of dogs, put Hank. The figure illustrates this approach. The argument isn’t necessarily true, because you know that not all dogs have fleas. All this shows is that the argument is valid. If the two premises are true, then the conclusion must be true. Now consider an argument involving rectangles and triangles. A polygon is a figure made up of line segments connected at their endpoints. When analyzing the validity of this argument, the Euler diagram starts with a circle containing all polygons, as shown here. Two circles are drawn inside the larger circle—one containing rectangles and the other triangles. The two circles don’t overlap, because rectangles have four sides, and triangles have three sides. The argument is invalid. Rectangles are not triangles—not even sometimes. Arguments can have more than two premises. For example: One Euler diagram that can represent this situation has three intersecting circles, as shown here. As you can see from the diagram, there can be presidents born in Kentucky who were not lawyers in Illinois and there can be presidents who were lawyers in Illinois but not born in Kentucky. The argument is invalid. To be valid, it must always be true.

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Calculation & Analysis How to Analyze Compound Statements

Article / Updated 07-30-2018

If your finite math instructor asks you to analyze a compound statement, you can try using a truth table to do this. Not every topic in a discussion can be turned into a compound statement and analyzed for its truth that way, but using logic and truth values is a good technique to use when possible. Consider the compound statement When constructing a truth table, you start with the basic p and q columns. Then you add a ~ q column followed by a column Before you can perform the conjunction, ^, you need a ~ p column. Here’s a step-by-step procedure. Start with a basic p and q and then add ~ q. When adding the column, perform the disjunction on the first and third columns. Remember, with disjunctions, the statement is false only when both component statements are false. Add the ~ p column. Add the column, which shows the conjunction of the fourth and fifth columns. The conjunction is true only when the two component statements are true. This complex statement is only true when both original statements are false.

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