AP Statistics: Sampling Distributions

This chapter introduces the concept of a sampling distribution, which is the probability distribution of a statistic (like a sample mean or proportion) obtained from a large number of samples of the same size from a population. It forms the theoretical foundation for making inferences about a population based on a single sample. You will learn how to describe the shape, center, and spread of the sampling distributions for sample proportions and sample means.

The main sub-topics are the sampling distribution of a sample proportion (for categorical data) and the sampling distribution of a sample mean (for quantitative data). A significant part of the chapter is dedicated to the Central Limit Theorem (CLT), which states that the sampling distribution of the sample mean will be approximately Normal if the sample size is large enough, regardless of the population's shape. You will also study the distributions for the difference between two sample proportions or two sample means.

The most common difficulty is understanding the abstract idea that a statistic is a random variable with its own distribution. This chapter relies heavily on concepts from probability and random variables. Students must be precise in checking the conditions for each scenario, such as the 10% condition for independence and the Large Counts or Normal/Large Sample conditions for assuming Normality, as these are critical for all future inference procedures.

Formulas for AP Statistics: Sampling Distributions

Mean of the sampling distribution of a sample proportion: $\mu_{\hat{p}} = p$ p is the true population proportion.
Standard deviation of the sampling distribution of a sample proportion: $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ Requires the 10% condition: the sample size n is no more than 10% of the population size N.
Large Counts Condition for sample proportions: $np \ge 10 \text{ and } n(1-p) \ge 10$ Condition to assume the sampling distribution of p-hat is approximately Normal.
Mean of the sampling distribution of a sample mean: $\mu_{\bar{x}} = \mu$ μ is the true population mean.
Standard deviation of the sampling distribution of a sample mean: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$ σ is the population standard deviation. Requires the 10% condition (n ≤ 0.10N).
Central Limit Theorem (CLT) condition: $n \ge 30$ If the population is not Normal, the sampling distribution of x-bar is approximately Normal if n is large enough.
Mean of the sampling distribution of a difference in proportions: $\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$ For two independent random samples.
Standard deviation of the sampling distribution of a difference in proportions: $\sigma_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}$ Requires the 10% condition for both independent samples.
Mean of the sampling distribution of a difference in means: $\mu_{\bar{x}_1 - \bar{x}_2} = \mu_1 - \mu_2$ For two independent random samples.
Standard deviation of the sampling distribution of a difference in means: $\sigma_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$ Requires the 10% condition for both independent samples.