AP Statistics: Inference for Proportions and Means

This chapter introduces statistical inference, the process of using sample data to draw conclusions about a larger population. You will learn the two primary methods of inference: confidence intervals, which provide a range of plausible values for a population parameter, and significance tests (or hypothesis tests), which assess the evidence for a claim about that parameter. The goal is to move beyond simply describing data and begin making quantified claims about the real world from which the data came.

The chapter is structured around the type of data being analyzed. You will learn to construct intervals and perform tests for a single population proportion, the difference between two population proportions, a single population mean, and the difference between two population means. This also includes procedures for paired data, which is a special case of inference for means.

The most common difficulty is choosing the correct inferential procedure for a given scenario and verifying the necessary conditions. Each test has specific requirements for randomness, independence, and normality that must be met and justified. This chapter builds directly on the concepts of probability and sampling distributions, so a firm grasp of those topics is essential for understanding why these methods work.

Formulas for AP Statistics: Inference for Proportions and Means

One-Sample z-Interval for a Proportion: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ z* is the critical value for the confidence level; \hat{p} is the sample proportion.
One-Sample z-Test Statistic for a Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ p_0 is the hypothesized population proportion from the null hypothesis.
Two-Sample z-Interval for a Difference in Proportions: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$ Used to estimate the difference between two population proportions, p_1 - p_2.
Two-Sample z-Test Statistic for a Difference in Proportions: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)(\frac{1}{n_1} + \frac{1}{n_2})}}$ \hat{p}_c is the pooled sample proportion, used when H₀: p₁ = p₂.
One-Sample t-Interval for a Mean: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$ t* is the critical value from a t-distribution with n-1 degrees of freedom.
One-Sample t-Test Statistic for a Mean: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ \mu_0 is the hypothesized population mean from the null hypothesis.
Two-Sample t-Interval for a Difference in Means: $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$ Used to estimate the difference between two population means, \mu_1 - \mu_2.
Two-Sample t-Test Statistic for a Difference in Means: $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ Used to test the hypothesis H₀: \mu₁ = \mu₂.
Paired t-Test Statistic: $t = \frac{\bar{x}_d - \mu_d}{s_d/\sqrt{n}}$ Tests the mean difference \bar{x}_d for paired data; often \mu_d = 0.