# 11: The Chi-Square Distribution

A chi-squared test is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true.

• 11.1: Prelude to The Chi-Square Distribution
You will now study a new distribution, one that is used to determine the answers to such questions. This distribution is called the chi-square distribution.
• 11.2: Facts About the Chi-Square Distribution
The chi-square distribution is a useful tool for assessment in a series of problem categories. These problem categories include primarily (i) whether a data set fits a particular distribution, (ii) whether the distributions of two populations are the same, (iii) whether two events might be independent, and (iv) whether there is a different variability than expected within a population.
• 11.3: Goodness-of-Fit Test
In this type of hypothesis test, you determine whether the data "fit" a particular distribution or not. For example, you may suspect your unknown data fit a binomial distribution. You use a chi-square test (meaning the distribution for the hypothesis test is chi-square) to determine if there is a fit or not. The null and the alternative hypotheses for this test may be written in sentences or may be stated as equations or inequalities.
• 11.4: Test of Independence
Tests of independence involve using a contingency table of observed (data) values. The test statistic for a test of independence is similar to that of a goodness-of-fit test.
• 11.5: Test for Homogeneity
The goodness–of–fit test can be used to decide whether a population fits a given distribution, but it will not suffice to decide whether two populations follow the same unknown distribution. A different test, called the test for homogeneity, can be used to draw a conclusion about whether two populations have the same distribution. To calculate the test statistic for a test for homogeneity, follow the same procedure as with the test of independence.
• 11.6: Comparison of the Chi-Square Tests
You have seen the Chi-square test statistic used in three different circumstances. The following bulleted list is a summary that will help you decide which Chi-square test is the appropriate one to use.
• 11.7: Test of a Single Variance (Not Included in the Course)
A test of a single variance assumes that the underlying distribution is normal. The null and alternative hypotheses are stated in terms of the population variance (or population standard deviation). A test of a single variance may be right-tailed, left-tailed, or two-tailed
• 11.8: Lab 1: Chi-Square Goodness-of-Fit (Worksheet)
A statistics Worksheet: The student will evaluate data collected to determine if they fit either the uniform or exponential distributions.
• 11.9: Lab 2: Chi-Square Test of Independence (Worksheet)
A statistics Worksheet: The student will evaluate if there is a significant relationship between favorite type of snack and gender.
• 11.E: The Chi-Square Distribution (Exercises)
These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.