Inference for Categorical Data: Proportions

Notes

Confidence intervals for $p$

Point estimate $\pm$ margin of error:

• Conditions: random sample, 10% condition ($n \le 10\%$ of the population), large counts ($n\hat p \ge 10$ and $n(1 - \hat p) \ge 10$).
• $z^* = 1.645$ for 90%, $1.96$ for 95%, $2.576$ for 99%.
• Interpretation: "We are 95% confident that the interval from ___ to ___ captures the true proportion of [context]."
• Confidence level meaning: if we took many samples and built an interval from each, about 95% of them would capture $p$.
• Larger $n$ makes the margin of error smaller; higher confidence makes it bigger.

Significance tests for $p$

1. State: $H_0: p = p_0$ and $H_a: p >, <, \ne p_0$, with $\alpha$ and a definition of $p$. 2. Plan: one-sample $z$ test; check conditions using $p_0$. 3. Do: $\displaystyle z = \frac{\hat p - p_0}{\sqrt{p_0(1 - p_0)/n}}$ and the p-value. 4. Conclude: if p-value $< \alpha$, reject $H_0$; there is convincing evidence for $H_a$. Otherwise fail to reject.

• p-value: the probability of getting a result at least as extreme as the one observed, assuming $H_0$ is true.
• Type I error: reject a true $H_0$ (probability $\alpha$). Type II error: fail to reject a false $H_0$. Power $= 1 - P(\text{Type II})$; it goes up with larger $n$ or larger $\alpha$.

Two proportions

• Interval: $\displaystyle (\hat p_1 - \hat p_2) \pm z^*\sqrt{\frac{\hat p_1(1 - \hat p_1)}{n_1} + \frac{\hat p_2(1 - \hat p_2)}{n_2}}$.
• Test of $H_0: p_1 = p_2$ uses the pooled proportion $\displaystyle \hat p_c = \frac{x_1 + x_2}{n_1 + n_2}$.
• If an interval for $p_1 - p_2$ contains 0, there's no convincing evidence of a difference.

Chi-square tests for two-way tables

• Homogeneity: several populations or treatments, one categorical variable. Independence: one sample, two categorical variables.
• Expected count $\displaystyle = \frac{\text{row total} \times \text{column total}}{\text{table total}}$; all expected counts should be at least 5.
• $\displaystyle \chi^2 = \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}$ with $df = (\text{rows} - 1)(\text{columns} - 1)$.