Chi-Square Tests

Overview

Chi-square ($\chi^2$) tests are used for categorical data. There are three types:

| Test | Purpose | |------|---------| | Goodness of Fit | Does a distribution match a claimed distribution? | | Homogeneity | Do different populations have the same distribution? | | Independence | Are two categorical variables independent? |

Chi-Square Test Statistic

$\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$

Always calculated the same way for all three tests.

1. Goodness of Fit Test

Purpose: Test whether a categorical variable follows a specified distribution.

Hypotheses:

• $H_0$: The data follows the specified distribution
• $H_a$: The data does not follow the specified distribution

Degrees of freedom: $df = k - 1$ (where $k$ = number of categories)

Expected counts: $E_i = n \cdot p_i$ (sample size × hypothesized proportion)

2. Test for Homogeneity

Purpose: Test whether the distribution of a categorical variable is the same across different populations.

Hypotheses:

• $H_0$: The distributions are the same for all populations
• $H_a$: The distributions are not all the same

Degrees of freedom: $df = (r-1)(c-1)$

Expected counts: $E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}$

3. Test for Independence

Purpose: Test whether two categorical variables are independent within a single population.

Hypotheses:

• $H_0$: The two variables are independent
• $H_a$: The two variables are not independent

Degrees of freedom: $df = (r-1)(c-1)$

Expected counts: Same formula as homogeneity

Conditions for All Chi-Square Tests

1. Random: Data from a random sample or randomized experiment
2. 10%: $n < 0.10N$
3. Large Counts: All expected counts $\geq 5$

Properties of the Chi-Square Distribution

• Always $\geq 0$
• Right-skewed (becomes less skewed as $df$ increases)
• P-value is always from the right tail
• Different shape for each $df$

Follow-Up Analysis

If you reject $H_0$, identify which cells contribute most to $\chi^2$ by examining: $\frac{(O - E)^2}{E}$ for each cell. Large contributions indicate where the biggest discrepancies are.

AP Tip: Chi-square tests are ALWAYS right-tailed. There is no "left-tailed" or "two-tailed" chi-square test. Show expected counts and verify they are all ≥ 5.