Statistics: 1001 Practice Problems For Dummies (+ Free Online Practice)
Overview
Become more likely to succeed—gain stats mastery with Dummies
Statistics: 1001 Practice Problems For Dummies gives you 1,001 opportunities to practice solving problems from all the major topics covered in Statistics classes—in the book and online! Get extra help with tricky subjects, solidify what you’ve already learned, and get in-depth walk-throughs for every problem with this useful book. These practice problems and detailed answer explanations will help you gain a valuable working knowledge of statistics, no matter what your skill level. Thanks to Dummies, you have a resource to help you put key stats concepts into practice.
- Work through practice problems on all Statistics topics covered in school classes
- Read through detailed
explanations of the answers to build your understanding
The material presented in Statistics: 1001 Practice Problems For Dummies is an excellent resource for students, as well as parents and tutors looking to help supplement Statistics instruction.
Statistics: 1001 Practice Problems For Dummies (9781119883593) was previously published as 1,001 Statistics Practice Problems For Dummies (9781118776049). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.
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statistics: 1001 practice problems for dummies (+ free online practice)
CHEAT SHEET
There are many types of statistics problems, including the use of pie charts, bar graphs, means, standard deviation to correlation, regression, confidence intervals, and hypothesis tests.To be successful, you need to be able to make connections between statistical ideas and statistical formulas. Through practice, you see what type of technique is required for a problem and why, as well as how to set up the problem, work it out, and make proper conclusions.
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Articles from
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When solving statistics problems, you must know the ways to find binomial probabilities. In these practice questions, pay special attention to the normal approximation. Solve the following problems about the basics of binomial random variables.
Sample questions
You roll a six-faced die ten times and record which face comes up each time (X).
Formulas abound in statistics problems — there's just no getting around them. However, there's typically a method to the madness if you can break the formulas into pieces. Here are some helpful tips:
Formulas for descriptive statistics basically take the values in the data set and apply arithmetic operations.
Statistics
Calculating a confidence interval for a population mean when the population standard deviation is known and the sample size is at least 30 involves the Z-distribution. When the population standard deviation is unknown, it involves a t-distribution. Calculate confidence intervals for population means in the following problems.
Symbols (or notation) found in statistics problems fall into three categories: math symbols, symbols referring to a population, and symbols referring to a sample. Math symbols are easy enough to decipher with a simple review of algebra; they involve items such as square root signs, equations of a line, and combinations of math operations.
Statistics
These practice problems focus on distinguishing discrete versus continuous random variables. Random variables represent quantities or qualities that randomly change within a population. Solve the following problems about discrete and continuous random variables.
Sample questions
Which of the following random variables is discrete?
When interpreting graphs in statistics, you might find yourself having to compare two or more graphs. The following histograms represent the grades on a common final exam from two different sections of the same university calculus class.
Credit: Illustration by Ryan Sneed
Credit: Illustration by Ryan Sneed
Sample questions
How would you describe the distributions of grades in these two sections?
The sample statistics questions here require that you compare three box plots. These side-by-side box plots represent home sale prices (in thousands of dollars) in three cities in 2012.
Credit: Illustration by Ryan Sneed
Sample questions
From high to low, what is the order of the cities' median home sale prices?
When working on statistics problems, you probably will have occasion to compare two box plots. The following box plots represent GPAs of students from two different colleges, call them College 1 and College 2.
Credit: Illustration by Ryan Sneed
Sample questions
What information is missing on this graph and on the box plots?
Most hypothesis tests use a similar framework, whether you are testing one population mean or the difference between two population means, some patterns will develop, but each hypothesis test has its own special elements.
Sample questions
A manager of a large grocery store chain believes that happy employees are more productive than unhappy ones.
A confidence interval is a range of likely values for the population parameter. Suppose that you want to find the value of a certain population parameter (for example, the average gas price in Ohio). If the population is too large, you take a sample (such as 100 gas stations chosen at random) and use those results to estimate the population parameter.
Statistics
Many real-world scenarios are looking to compare two populations. For example, what is the difference in survival rates for cancer patients taking a new drug compared to cancer patients on the existing drug? What is the difference in the average salary for males versus females? What’s the difference in the average price of gas this year compared to last year?
Statistics
You can use probabilities from a two-way table to look for and describe relationships between two categorical variables. The following table displays information about cigarette smoking and diagnosis with hypertension for a group of patients at a medical clinic.
Credit: Illustration by Ryan Sneed
The following stacked bar graph displays the smoking and hypertension data for the group of patients at the medical clinic.
Anytime you're trying to estimate a number from a population (like the average gas price in the United States), you include a margin of error. You need to accurately calculate the sample size needed to get a particular margin of error you want.
For the three practice questions here, consider that:
A sociologist is interested in the average age women get married.
Statistics
Sampling distributions and the central limit theorem are difficult topics. Everything builds on the understanding of what a sampling distribution is — work until you get it. Solve the following problems about sample means and sampling distributions.
Sample questions
A researcher draws a series of samples of exam scores from a population of scores that has a normal distribution.
Statistics
In the following practice problems, you will be connecting a population proportion to a survey. A website ran a random survey of 200 customers who purchased products online in the past 12 months. The survey found that 150 customers were "very satisfied."
Sample questions
What are the sample proportion and the standard error for the sample proportion, based on this data?
Statistics
Your job here is to find and interpret the results of a regression line and its elements and to carefully check exactly how well your line fits. Note: Regression assumes you've found that a strong relationship exists.
Use the following scatter plot to answer the following problems.
Credit: Illustration by Ryan Sneed
Sample questions
In terms of numbers, what is the most plausible value for the correlation between X and Y?
Statistics
Be aware of the units of any descriptive statistic you calculate (for example, dollars, feet, or miles per gallon). Some descriptive statistics are in the same units as the data, and some aren't. Solve the following problems about data sets and descriptive statistics.
Sample questions
Which of the following descriptive statistics is least affected by adding an outlier to a data set?
A margin of error is the "plus or minus" part you have to add to your statistical results to tell everyone you acknowledge that sample results will vary from sample to sample, and could vary from the actual population condition. The margin of error helps you indicate how much you believe those results could vary, with a certain level of confidence.
Statistics
The normal distribution is the most common distribution of all. Its values take on that familiar bell shape, with more values near the center and fewer as you move away. Solve the following problems about the definition of the normal distribution and what it looks like.
Sample questions
What are properties of the normal distribution?
Statistics
When working with random variables, you need to be able to calculate and interpret the mean. For these problems, let X be the number of classes taken by a college student in a semester. Use the formula for the mean of a discrete random variable X to answer the following problems:
Sample questions
If 40% of all the students are taking four classes, and 60% of all the students are taking three classes, what is the mean (average) number of classes taken for this group of students?
The practice questions here help you in determining appropriate sample sizes required to achieve a certain margin of error. Figure out the sample size needed in the following problems.
Sample questions
A physician wants to estimate the average BMI (body mass index, a measure that combines information about height and weight) for her adult patients.
Statistics
The empirical rule in statistics states that for a normal distribution, almost all data will fall within three standard deviations of the mean. Use the empirical rule to solve the following problems.
Sample questions
According to the empirical rule (or the 68-95-99.7 rule), if a population has a normal distribution, approximately what percentage of values is within one standard deviation of the mean?
Statistics
For the sample questions here, X is a random variable with a binomial distribution with n = 11 and p = 0.4. Use the binomial table to answer the following problems.
Sample questions
What is P(X = 5)?
Answer: 0.221
The binomial table has a series of mini-tables inside of it, one for each selected value of n. To find P(X = 5), where n = 11 and p = 0.
Statistics
You might need to references the Z-table to solve the following questions.
Sample questions
For the following questions, consider that in a population of adults ages 18 to 65, BMI (body mass index) is normally distributed with a mean of 27 and a standard deviation of 5.
What is the BMI score for which half of the population has a lower value?
Statistics
Solving statistics problems can involve finding probabilities, mean, and standard deviation for a specific random variable, in this case the binomial. Solve the following problems about the mean, standard deviation, and variance of binomial random variables.
Sample questions
What is the mean of a binomial random variable with n = 18 and p = 0.
Statistics
The three practice questions here can help you achieve an understanding of the relationship between values of z and the confidence level needed for a margin of error. Use the following table to find the appropriate z*-value for the confidence levels given except where noted.
Sample questions
The Z-table and the preceding table are related but not the same.
Statistics
Here, you get to practice finding binomial probabilities by using a formula. The following problems have a binomial random variable with p = 0.55. Use the following formulas for the binomial distribution for the problems.
where
and
n! = (n – 1)(n – 2)(n – 3) . . . (3)(2)(1)
Sample questions
What is the probability of exactly one success in eight trials?
Statistics
"Man does not live by a correlation alone." You always need to look at a scatter plot of the data as well. (Neither is foolproof by itself.) You conduct a study to see whether the amount of time spent studying per week is related to GPA for a group of college computer science majors.
Sample questions
How do you designate the "time spent studying" variable on a scatter plot of your data?
Statistics
As you work with statistics, you will encounter some vocabulary terms, such as bias, variables, and the mean that you'll need to understand to answer problems. As soon as you understand what the language means, you immediately will start feeling more comfortable.
Bias is systematic favoritism in the data.
A variable is a characteristic or measurement on which data is collected and whose result can change from one individual to the next.
Statistics
When working with statistics, it's important to understand some of the terminology used, including quantitative and categorical variables and how they differ. The trick is to get a handle on the lingo right from the get-go, so when it comes time to work the problems, you’ll pick up on cues from the wording and get going in the right direction.
Statistics
When working on statistical problems, you sometimes will need to measure where a certain value stands in a data set by using percentiles. Creating a set of five numbers (using percentiles) can reveal some aspects of the shape, center, and variation in a data set.
Remember that a percentile isn't a percent, even though they sound the same!
Statistics
Everything's got its own lingo, and statistics is no exception. Some of the most common terms used in statistics include the population, sample, parameter, and statistic — also referred to as the big four. Here, you'll gain an understanding of these terms and the context in which they're used. You need to be able to pick out the big four in every situation; they'll follow you wherever you go.
Statistics
The t-table (for the t-distribution) is different from the z-table (for the z-distribution). Make sure you understand the values in the first and last rows. Finding probabilities for various t-distributions, using the t-table, is a valuable statistics skill.
How to use the t-table to find right-tail probabilities and p-values for hypothesis tests involving t:
First, find the t-value for which you want the right-tail probability (call it t), and find the sample size (for example, n).
You can use the z-score table to find a full set of "less-than" probabilities for a wide range of z-values using the z-score formula. Below you will find both the positive z-score and negative z-score table. The problem examples below the table will help you learn how to use it.
Z Score Table Sample Problems
Use these sample z-score math problems to help you learn the z-score formula.
When finding probabilities for a normal distribution (less than, greater than, or in between), you need to be able to write probability notations. Practice these skills by writing probability notations for the following problems.
Sample questions
Write the probability notation for the area shaded in this Z-distribution.
Numbers sitting in a little table seem easy enough, but you'd be surprised at all the information you can get out of a table, and how many equations, formulas, and notations that you can squeeze out of them. Solve the following problems about independent variables and two-way tables.
Sample questions
If variables A and B are independent, which of the following must be true?
A good graph displays data in a way that's fair, makes sense, and makes a point. Not all graphs possess these qualities. The following bar chart represents the post-graduation plans of the graduating seniors from one high school. Assume that every student chose exactly one of these five options.
(Note: A gap year means that the student is taking a year off before deciding what to do.
Box plots are a huge issue. Making a box plot itself is one thing; understanding the do's and (especially) the don'ts of interpreting box plots is a whole other story.
The following box plot represents data on the GPA of 500 students at a high school.
Credit: Illustration by Ryan Sneed
Sample questions
What is the range of GPAs in this data?
Pay special attention to how a histogram shows the variability in a data set. Flat histograms can have a lot of variability in the data, but flat time plots have none — that's one eye-opener. The following histogram represents the body mass index (BMI) of a sample of 101 U.S. adults.
Sample questions
Why are there no gaps between the bars of this histogram?
Graphs, in this case pie charts, should be able to stand alone and give all the information needed to identify the main point quickly and easily. The media gives the impression that making and interpreting graphs is no big deal. However, in statistics, you work with more complicated data, consequently taking your graphs up a notch.
Correlation is more than just a number; it's a way of describing relationships in a universal way. Correlation applies to quantitative variables (like age and height), even though the "street definition" of correlation relates any variables (like gender and voting pattern).
This scatter plot represents the high-school and freshman college GPAs of 24 students.
Time plots are used for highlighting data collected over time. The data in the following time chart shows the annual high-school dropout rate for a school system for the years 2001 to 2011.
Credit: Illustration by Ryan Sneed
Sample questions
What is the general pattern in the dropout rate from 2001 to 2011?
Answer: decreasing
Although not every year shows a decrease from the previous year, the clear overall pattern of the dropout rate is one of a steady decrease.
Conducting a hypothesis test is somewhat like doing detective work. Every population has a mean, and it's usually unknown. Many people claim that they know what it is; others assume that it hasn't changed from a past value; and in many cases, the population mean is supposed to follow certain specifications.
Your modus operandi is to challenge or test that value of the population mean that's already assumed, given, or specified and use data as your evidence.
Hypothesis testing is a scientific procedure for asking and answering questions. Hypothesis tests help people decide whether existing claims about a population are true, and they're also commonly used by researchers to see whether their ideas have enough evidence to be declared statistically significant.
Solve the following problems about using a hypothesis test.
With simple linear regression, you look for a certain type of relationship between two quantitative (numerical) variables (like high-school GPA and college GPA.) This special relationship is a linear relationship — one whose pairs of data resemble a straight line.
In this scatter plot, the two variables plotted are quantitative (numerical).
Statistics
The practice questions here have you working on margin of error for population means and population proportions. Solve the following problems related to margin of error and population proportion.
The following table provides the z*- values for selected (percentage) confidence levels.
Sample questions
A market researcher samples 100 people to find a confidence interval for estimating the average age of their customers.
Statistics
When you look at the difference between two means (or two proportions), keep track of what populations you're calling Population 1 and Population 2. Subtracting two numbers in the opposite order changes the sign of the results!
A random sample of 120 college students who were physics majors found that they spent an average of 25 hours a week on homework; the standard deviation for the population of physics majors was 7 hours.
It's easy to get caught up in all the calculations of regression. Always remember that understanding and interpreting your results is just as important as calculating them!
A building contractor examines the cost of having carpentry work done in some of his buildings in the current year. He finds that the cost for a given job can be predicted by this equation:
y = $50x + $65
Here, y is the cost of a job (in dollars), and x is the number of hours a job takes to complete.
Surveys are everywhere, and their quality can range from good to bad to ugly. The ability to critically evaluate surveys is important when you're working with statistics.
Sample questions
You're interested in the willingness of adult drivers (age 18 and over) in a metropolitan area to pay a toll to travel on less-congested roads.
Hypothesis testing can seem like a plug-and-chug operation, but that can take you only so far. Remember that a small p-value comes from a large test statistic, and both mean rejecting H0. Calculate p-values in the following problems.
Sample questions
A researcher has a less than alternative hypothesis and wants to run a single sample mean z-test.
Some graphs are easy to make and interpret, some are hard to make but easy to interpret, and some graphs are tricky to make and even trickier to interpret. But you can never get too much practice.
The following three histograms represent reported annual incomes, in thousands of dollars, from samples of 100 individuals from three professions; call the different incomes Income 1, Income 2, and Income 3.
Statistics
The problems presented here give you practice calculating and interpreting the mean, variance, and standard deviation of a random variable. In the following table, X represents the number of siblings for the 29 students in a first-grade class.
Sample questions
What is the mean number of siblings for these students?
The sample questions here focus on working with the sample means as a random variable. You have a six-sided die with faces numbered 1 through 6, and each face is equally likely to come up on any given roll.
Sample questions
What is the meaning for X in this example?
Answer: a random variable denoting the outcome from a single roll of the die
X is a random variable with possible values 1, 2, 3, 4, 5, and 6, denoting the outcome from a single roll of the die.
Two-way tables are used to organize and interpret pairs of categorical variables. Here, you practice reading and interpreting all parts of a two-way table. This 2-x-2 table displays results from a poll of randomly selected male and female college students at a certain college, asking whether they were in favor of increasing student fees to expand the college's athletics program.
Specific terms are used to describe problems with surveys and samples. Make sure you pick up on subtle differences. Here are the most common terms:
Sampling frame: A list of all the members of the target population.
Census: Getting desired information from everyone in the target population.
Random sample: Each member of the population has an equal chance of being selected for the sample.
In hypothesis testing, you might need to set up a pair of hypotheses: the current claim (null hypothesis) and the one challenging it (alternative hypothesis). Determine null and alternative hypotheses in the following problems.
Sample questions
You decide to test the published claim that 75% of voters in your town favor a particular school bond issue.
The problems here focus on calculating, interpreting, and comparing standard deviation and variance in basic statistics. Solve the following problems about standard deviation and variance.
Sample questions
What does the standard deviation measure?
Answer: how concentrated the data is around the mean
A standard deviation measures the amount of variability among the numbers in a data set.
Focus not only on the terms for the statistics and analyses you'll calculate but also on their interpretation, especially in the context of a statistics problem. Answer the following problems about different statistics and data analysis terms.
Sample questions
Which of the following data sets has a median of 3?
Statistics
There are many types of statistics problems, including the use of pie charts, bar graphs, means, standard deviation to correlation, regression, confidence intervals, and hypothesis tests.To be successful, you need to be able to make connections between statistical ideas and statistical formulas. Through practice, you see what type of technique is required for a problem and why, as well as how to set up the problem, work it out, and make proper conclusions.
Statistics
Solving statistics problems is always about having a strategy. You can't just read a problem over and over and expect to come up with an answer — all you'll get is anxiety! Although not all strategies work for everyone, here's a three-step strategy that has proven its worth:
Label everything the problem gives you.
Like every subject, statistics has its own language. The language is what helps you know what a problem is asking for, what results are needed, and how to describe and evaluate the results in a statistically correct manner. Here's an overview of the types of statistical terminology:
Four big terms in statistics are population, sample, parameter, and statistic:
A population is the entire group of individuals you want to study, and a sample is a subset of that group.
You can do hypothesis tests for three specific scenarios: testing one population proportion; testing for a difference between two population proportions; and testing for a difference between two population means. Solve the following problems about testing one population proportion. Note from JM: This isn't quite accurate.
The t-distribution is a relative of the normal distribution. It has a bell shape with values more spread out around the middle. That is, it's not as sharply curved as the normal distribution, which reflects its ability to work with problems that may not be exactly normal but are close.
Solve the following problems about the t-distribution, its traits, and how it compares to the Z-distribution.
Sample results vary — that's a major truth of statistics. You take a random sample of size 100, find the average, and repeat the process over and over with different samples of size 100. Those sample averages will differ, but the question is, by how much? And what affects the amount of difference?
Understanding this concept of variability between all possible samples helps determine how typical or atypical your particular result may be.
Statisticians need to understand the central limit theorem, how to use it, when to use it, and when it's not needed. The central limit theorem is used only in certain situations. Solve the following problems that involve the central limit theorem.
Sample questions
Suppose that a researcher draws random samples of size 20 from an unknown distribution.
Statistics
In the practice problems here, you will be finding probabilities for a random variable. The following table represents the probability distribution for X, the employment status of adults in a city.
Sample questions
If you select one adult at random from this community, what is the probability that the individual is employed part-time?
The wording of the problems for two-way tables can be extremely tricky; one small change in wording can lead to a totally different answer. Practice as many problems as you can.
Sample questions
Suppose that you have a 2-x-2 table displaying values on gender (male or female) and laptop computer ownership (yes or no) for 100 male and 100 female college students.
When you are doing hypothesis testing, you must be clear on Type I and Type II errors in the real sense — as false alarms and missed opportunities. Solve the following problems about Type I and Type II errors.
Sample questions
Which of the following describes a Type I error?
A. accepting the null hypothesis when it is true
B.
Statistics
Descriptive statistics are statistics that describe data. Two of the staple ingredients of descriptive statistics are the mean and median. Your first job in analyzing data is to identify, understand, and calculate these descriptive statistics. Solve the following problems about means and medians.
Sample questions
To the nearest tenth, what is the mean of the following data set?
Statistics
Use the t-table as needed and the following information to solve the following problems: The mean length for the population of all screws being produced by a certain factory is targeted to be
Assume that you don't know what the population standard deviation is. You draw a sample of 30 screws and calculate their mean length.
Statistics
For these three sample questions, consider that: A researcher conducted an Internet survey of 300 students at a particular college to estimate the average amount of money students spend on groceries per week. The researcher knows that the population standard deviation of weekly spending is $25. The mean of the sample is $85.
Many applications of statistics involve categorical variables, such as gender (male/female), opinion (yes/no/undecided), home ownership (yes/no), or blood type. One common statistical application is to look for relationships between two categorical variables.
Solve the following problems about variables in two-way tables.
When you explore linear relationships between a pair of quantitative (numerical) variables, X and Y, your basic question is this: As the X variable increases in value, does the Y variable increase with it, does it decrease in value, or does it just basically not react at all? The answer requires the use of graphs as well as the calculation and interpretation of a certain numerical measure of togetherness — correlation.
The practice problems here will help you understand the standard normal (Z-) distribution, its properties, and how its values are interpreted and used. In the problems below, the random variable X has a normal distribution, with a mean of 17 and a standard deviation of 3.5.
Sample questions
What is the z-score for a value of 21.
Statistics
Use the following formula for the variance of a discrete random variable X as needed to answer the following problems (round each answer to two decimal places):
Sample questions
If the variance of a discrete random variable X is 3, what is the standard deviation of X?
Answer: 1.73
The standard deviation is the square root of the variance, so if the variance of X is 3, the standard deviation of X is
(rounded to two decimal places).
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