Overview: Chi Square Chapter Notes | AP Statistics - Grade 9 PDF Download

Chi-Square Tests Two or More Categories
Students need to understand how to select from the following tests:

• Chi-Square Test for Goodness of Fit (for a distribution of proportions of one categorical variable in a population).
• Chi-Square Test for Independence (for associations between categorical variables within a single population).
• Chi-Square Test for Homogeneity (for comparing distributions of a categorical variable across populations or treatments).

To integrate conceptual understanding, students can make connections between frequency tables, conditional probability, and calculating expected counts. The chi-square statistic is introduced to measure the distance between observed and expected counts relative to expected counts.

Basis of Chi-Square Tests

A chi-square test is a statistical test that is used to determine whether there is a significant difference between the observed frequencies in a sample and the expected counts of a particular variable in a reference distribution. It is commonly used to test for associations between categorical variables.
For example, if one wants to analyze the difference between a person's state of residence and political party affiliation, a chi-square test could be done to compare the number of expected Democrat/Republican voters in a given state with the actual number of Democrat/Republican voters in that state. This difference would likely be significant in states such as California (mostly Democrat) and Alabama (mostly Republican). If this difference between actual and expected is great enough, we can have convincing evidence that these two variables are related.

What’s Needed?

In order to perform a hypothesis test using a chi-square procedure, one would need either a two-way table or frequency table distribution of our categorical variable(s). From there, we can compare our actual counts from the distribution to our expected counts based on a given probability.

Conditions

Just like we had with other inference procedures, our test hinges on certain conditions being met. With chi-square testing, we need the following two conditions:

• Our sample was taken randomly or treatments were assigned randomly in an experiment.
• Large Counts: All expected counts are at least 5. This is similar to our normal condition in previous inference procedures.

Example

In our voting example, Joe Biden received 51.3% of the vote nationwide in the 2020 elections, while Donald Trump garnered 46.9% of the vote. Based on these expected percentages, we would expect Joe Biden to receive about 1.2 million votes out of the approximate 2.3 million votes in Alabama. However, Joe Biden only received 849,000. Since there is such a discrepancy between our expected vote count and our actual vote count, we would likely conclude that state of residence and vote recipients are related in some way.

Test Taking Tip: Template to Use

When performing inference, it is a great idea to have a template that you follow to ensure you cover all bases when performing a free-response question (FRQ) on the exam. One popular inference template is SPDC:

• State (parameter of interest and hypotheses if necessary)
• Plan (Conditions for inference)
• Do (Calculations with calculator speak if using a calculator)
• Conclude (Conclusion based on interval or p-value)

This template is a huge test-taking tip that can help you be successful on the inference FRQ on the exam.

Key Terms to Review

• Chi-Square Test for Independence: A statistical method used to determine whether there is a significant association between two categorical variables.
• Chi-Square Statistic: A measure used to determine how well observed data fits an expected distribution.
• Chi-Square Tests: Statistical methods used to determine whether there is a significant association between categorical variables.
• Chi-Square Test for Homogeneity: A statistical method used to determine if different populations have the same distribution of a categorical variable.
• Conditional Probability: The likelihood of an event occurring given that another event has already occurred.
• Expected Counts: Predicted frequencies of occurrences in a contingency table under the assumption of independence.
• Frequency Tables: A way to organize and summarize data by showing how often each value or category occurs.
• Hypothesis Test: A method used to make inferences about a population based on sample data.
• One Proportion Z Intervals and Tests: Methods used to estimate and test hypotheses about a single population proportion.
• P-Value: A measure used in hypothesis testing to determine the strength of evidence against the null hypothesis.
• Random Sampling: A method of selecting individuals from a population so that every member has an equal chance of being chosen.
• Sampling Distribution: A probability distribution of a statistic obtained by selecting random samples from a population.
• Two Proportions Z Intervals and Tests: Methods used to compare the proportions of two different groups or populations.
• Two-Way Table: A statistical tool used to display the relationship between two categorical variables.