AP Statistics: Inference for Categorical and Quantitative Data

This chapter introduces statistical inference, which is the process of using sample data to draw conclusions about a larger population. You will learn to perform the two main types of inference: estimating a population parameter by creating a confidence interval, and assessing evidence for a claim about a parameter by performing a significance test. These procedures are applied to both categorical data (proportions) and quantitative data (means).

The main sub-topics include inference for a single proportion or mean, comparing two proportions or means, and inference for paired data. You will also cover chi-square tests for categorical data (goodness of fit, homogeneity, and independence) and inference for the slope of a least-squares regression line. By the end, you should be able to identify the correct inference procedure for a given scenario, verify the necessary conditions, perform the calculations, and interpret the results in context.

The most common difficulty is choosing the correct statistical test from the many options available. For example, students often struggle to distinguish between a two-sample t-test and a paired t-test, or between different types of chi-square tests. This chapter builds directly on the concepts of sampling distributions and probability, so a strong understanding of the Central Limit Theorem is essential.

Formulas for AP Statistics: Inference for Categorical and Quantitative Data

Confidence Interval for a Population Proportion: $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ Where \hat{p} is the sample proportion and z* is the critical value for the confidence level.
Test Statistic for a One-Sample z-Test for a Proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ Where p_0 is the hypothesized population proportion.
Confidence Interval for a Difference in Two Proportions: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$ For two independent random samples.
Test Statistic for a Two-Sample z-Test for Proportions: $z = \frac{(\hat{p}_1 - \hat{p}_2) - 0}{\sqrt{\hat{p}_c(1-\hat{p}_c)(\frac{1}{n_1} + \frac{1}{n_2})}}$ Where \hat{p}_c is the pooled sample proportion, used when H₀ is p₁ = p₂.
Confidence Interval for a Population Mean: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$ Where \bar{x} is the sample mean, s is the sample standard deviation, and t* has n-1 degrees of freedom.
Test Statistic for a One-Sample t-Test for a Mean: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ Where \mu_0 is the hypothesized population mean. Degrees of freedom are n-1.
Test Statistic for a Two-Sample t-Test for Means: $t = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ For comparing the means of two independent groups.
Test Statistic for a Paired t-Test: $t = \frac{\bar{x}_d - \mu_d}{s_d/\sqrt{n}}$ The subscript d refers to the differences between paired data points; \mu_d is often 0.
Chi-Square Test Statistic: $\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}$ Used for goodness of fit, homogeneity, and independence tests.
Confidence Interval for the Slope of a Regression Line: $b \pm t^* \cdot SE_b$ Where b is the sample slope and SE_b is its standard error.
Test Statistic for the Slope of a Regression Line: $t = \frac{b - \beta_0}{SE_b}$ Tests if a linear relationship exists. The null hypothesis is usually \beta_0 = 0.