Chi-Square Tests

Chi-Square Test Statistic

$\chi^2 = \sum \frac{(O - E)^2}{E}$ where $O$ = observed count, $E$ = expected count.

1. Goodness-of-Fit Test

• Purpose: Does the distribution of one categorical variable match a claimed distribution?
• $H_0$: The distribution matches the claimed proportions.
• $df = k - 1$ where $k$ = number of categories.
• Expected: $E_i = n \cdot p_i$.

2. Test of Independence

• Purpose: Are two categorical variables independent (in a single population)?
• $H_0$: The variables are independent.
• Data in a two-way table.
• $df = (r-1)(c-1)$.
• Expected: $E = \frac{\text{row total} \times \text{column total}}{n}$.

3. Test of Homogeneity

• Purpose: Do different populations have the same distribution of a categorical variable?
• Same mechanics as the test of independence.
• $H_0$: The distribution is the same across populations.

Conditions

1. Random sample(s).
2. All expected counts $\geq 5$.
3. Independence (10% condition).

Properties of $\chi^2$

• Always $\geq 0$.
• Right-skewed distribution.
• Larger $\chi^2$ → more evidence against $H_0$.
• Always a right-tail test.

Problem: A die is rolled 60 times. Results: 8, 12, 10, 9, 11, 10 for faces 1-6. Test if the die is fair ($\alpha = 0.05$).

Solution:

$H_0$: The die is fair (each face has probability 1/6). $E = 60/6 = 10$ for each face.

$\chi^2 = \frac{(8-10)^2}{10} + \frac{(12-10)^2}{10} + \frac{(10-10)^2}{10} + \frac{(9-10)^2}{10} + \frac{(11-10)^2}{10} + \frac{(10-10)^2}{10}$ $= 0.4 + 0.4 + 0 + 0.1 + 0.1 + 0 = 1.0$

$df = 5$. $\chi^2_{\text{crit}} = 11.07$ at $\alpha = 0.05$. Since $1.0 < 11.07$, fail to reject $H_0$. No evidence the die is unfair.

• $\chi^2 = \sum (O-E)^2/E$ for all cells.
• Goodness-of-fit: one variable, known distribution.
• Independence/Homogeneity: two-way tables; same calculation, different context.
• Conditions: random, expected counts ≥ 5, independence.
• Always a right-tail test.