Key Takeaways: Inference for Proportions

One-Sentence Summary

The one-sample z-test for proportions uses the sampling distribution of $\hat{p}$ (grounded in the CLT from Chapter 11) to test claims about population proportions — with the success-failure condition as the gatekeeper, $p_0$ in the test's standard error, and the Wilson interval as an improvement over the basic Wald CI for challenging situations.

Core Concepts at a Glance

The One-Sample Z-Test for a Proportion

The Five Steps

The Test Statistic

$$\boxed{z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}}$$

Uses $p_0$ in the SE because we compute under the assumption that $H_0$ is true.

The Conditions

Confidence Intervals: Three Methods

Key difference from hypothesis tests: CIs use $\hat{p}$ in the SE (not $p_0$) because there's no hypothesized value.

Margin of Error in Polling

$$E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Critical caveat: The margin of error measures random sampling error only — not bias (nonresponse, coverage, question wording). A biased poll can have a small margin of error and still be very wrong.

The $p_0$ vs. $\hat{p}$ Rule

Common Misconceptions

Python Quick Reference

import numpy as np
from scipy import stats
from statsmodels.stats.proportion import proportions_ztest, proportion_confint

# --- One-sample z-test for a proportion ---
z_stat, p_value = proportions_ztest(count=208, nobs=400,
                                     value=0.47,
                                     alternative='larger')

# --- Confidence intervals (three methods) ---
ci_wald = proportion_confint(208, 400, alpha=0.05, method='normal')
ci_wilson = proportion_confint(208, 400, alpha=0.05, method='wilson')

# --- Plus-four (manual) ---
p_tilde = (208 + 2) / (400 + 4)
se_tilde = np.sqrt(p_tilde * (1 - p_tilde) / (400 + 4))
ci_pf = (p_tilde - 1.96 * se_tilde, p_tilde + 1.96 * se_tilde)

# --- Exact binomial test (when success-failure fails) ---
result = stats.binomtest(k=3, n=20, p=0.05, alternative='greater')

How This Chapter Connects

Threads Updated

• Sam and Daria: Formal proportion test conducted ($z = 1.30$, $p = 0.097$). Still fail to reject at $\alpha = 0.05$ with only 65 shots. Need more data — power analysis in Ch.17.
• Maya: Tested county hypertension prevalence above national rate — found significant evidence ($z = 2.00$, $p = 0.023$). Used proportion inference to estimate disease prevalence for Bayesian screening analysis.
• Polling and elections: Margin of error demystified. Random error vs. bias distinction. The 2016/2020 lessons.

The One Thing to Remember

If you forget everything else from this chapter, remember this:

In a hypothesis test, the standard error uses $p_0$ (because you're computing under the null). In a confidence interval, the standard error uses $\hat{p}$ (because there's no null to assume). The success-failure condition is your checkpoint for whether the normal approximation is valid. And the margin of error in a poll only measures random sampling error — not bias, not nonresponse, not question wording. The uncertainty in any proportion estimate is always at least as large as the margin of error, and usually larger.

Key Terms