Sampling distribution of a difference in sample means

What the exam asks

Expect tasks using these AP task verbs:

• Calculate and interpret the mean $\mu_{\bar{x}_1 - \bar{x}_2}$ and standard deviation $\sigma_{\bar{x}_1 - \bar{x}_2}$ in context.
• Justify the randomization, 10%, and normality conditions for two independent groups.
• Describe the sampling distribution of $\bar{x}_1 - \bar{x}_2$ (shape, center, and variability).
• Calculate and interpret the probability of obtaining a sample mean difference at least as extreme as a given value using the normal distribution.

Core ideas

Parameters of the sampling distribution

Suppose we have two independent populations (or two treatment groups in a randomized experiment) with means $\mu_1, \mu_2$ and standard deviations $\sigma_1, \sigma_2$. For independent samples of sizes and , the sampling distribution of has:

Conditions for a valid model

Every condition must be verified for both groups in context: