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### How to Find Percentiles for a *t*-Distribution

When you want to find percentiles for a *t*-distribution, you can use the *t*-table. A *percentile* is a number on a statistical distribution whose less-than probability is the given percentage; for example, [more…]

### How to Find *t**-Values for Confidence Intervals

*Confidence intervals* estimate population parameters, such as the population mean, by using a statistic (for example, the sample mean) plus or minus a margin of error. To compute the margin of error for [more…]

### How to Identify a Sampling Distribution

In statistics, a sampling distribution is based on sample averages rather than individual outcomes. This makes it different from a distribution. Here’s why: A [more…]

### How Sample Size Affects Standard Error

The size (*n*) of a statistical sample affects the standard error for that sample. Because *n* is in the denominator of the standard error formula, the standard error decreases as [more…]

### How Population Standard Deviation Affects Standard Error

In statistics, the standard deviation in a population affects the standard error for that population. Standard deviation measures the amount of variation in a population. In the standard error formula [more…]

### How a Normal Distribution Affects the Shape of a Sampling Distribution

In statistics, when the original distribution for a population *X* is normal, then you can also assume that the shape of the sampling distribution, or [more…]

### How a Sampling Distribution Is Affected When the Distribution Is Not Normal

In statistics, if a population *X* has any distribution that is *not* normal, or if its distribution is unknown, you can’t automatically say the distribution of the sample means [more…]

### How to Find the Sampling Distribution of a Sample Proportion

If you use a large enough statistical sample size, you can apply the Central Limit Theorem (CLT) to a sample proportion for categorical data to find its sampling distribution. The [more…]

### How to Find Probabilities for a Sample Proportion

You can find probabilities for a sample proportion by using the normal approximation as long as certain conditions are met. For example, say that a statistical study claims that 0.38 or 38% of all the [more…]

### What the Margin of Error Tells You About a Statistical Sample

If you read statistical survey results without knowing the margin of error, or MOE, you are only getting part of the story. Survey results themselves (with no MOE) are only a measure of how the [more…]

### How to Calculate the Margin of Error for a Sample Proportion

When you report the results of a statistical survey, you need to include the margin of error. The general formula for the margin of error for a sample proportion [more…]

### How Sample Size Affects the Margin of Error

In statistics, the two most important ideas regarding sample size and margin of error are, first, sample size and margin of error have an inverse relationship; and second, after a point, increasing the [more…]

### Choosing a Confidence Level for a Population Sample

In statistics, every confidence interval (and every margin of error, for that matter) has a percentage associated with it, called a *confidence level**.* This percentage represents how confident you are that [more…]

### How to Determine the Minimum Size Needed for a Statistical Sample

The margin of error of a confidence interval (CI) is affected by size of the statistical sample; as the size increases, margin of error decreases. Looking at this the other way around, if you want a smaller [more…]

### Creating a Confidence Interval for the Difference of Two Means with Known Standard Deviations

If you know the standard deviations for two population samples, then you can find a confidence interval (CI) for the difference between their means, or averages. The goal of many statistical surveys and [more…]

### How to Create a Confidence Interval for the Difference of Two Means with Unknown Standard Deviations and/or Small Sample Sizes

You can find a confidence interval (CI) for the difference between the means, or averages, of two population samples, even if the population standard deviations are unknown and/or the sample sizes are [more…]

### How to Estimate the Difference between Two Proportions

To estimate the difference between two population proportions with a confidence interval, you can use the Central Limit Theorem when the sample sizes are large enough [more…]

### How to Find the Cutoff Point for Rejecting a Null Hypothesis

In statistics, if you want to draw conclusions about a null hypothesis H_{0} (reject or fail to reject) based on a *p-*value, you need to set a predetermined cutoff point where only those [more…]

### How to Test a Hypothesis for One Population Mean

You can use a hypothesis test to examine or challenge a statistical claim about a population mean if the variable is numerical (for example, age, income, time, and so on) and only one population or group [more…]

### How to Use the *t*-Test to Handle Small Samples and Unknown Standard Deviations

When using a test statistic for one population mean, there are two cases where you must use the *t*-distribution instead of the *Z*-distribution. The first case is where the sample size is small [more…]

### How to Test a Null Hypothesis Based on One Population Proportion

You can use a hypothesis test to test a statistical claim about a population proportion when the variable is categorical (for example, gender or support/oppose) and only one population or group is being [more…]

### How to Compare Two Independent Population Averages

You can compare numerical data for two statistical populations or groups (such as cholesterol levels in men versus women, or income levels for high school versus college grads) to test a claim about the [more…]

### How to Test for an Average Difference Using the Paired *t*-Test

You can test for an average difference using the paired *t*-test when the variable is numerical (for example, income, cholesterol level, or miles per gallon) and the individuals in the statistical sample [more…]

### How to Calculate a Correlation

Can one statistic measure both the strength and direction of a linear relationship between two variables? Sure! Statisticians use the *correlation coefficient* [more…]

### How to Interpret a Correlation Coefficient *r*

In statistics, the correlation coefficient *r* measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of [more…]