Sampling distribution of a difference in sample proportions

What the exam asks

You will be expected to:

• Calculate the mean and standard deviation of the sampling distribution of $\hat{p}_1 - \hat{p}_2$ given population proportions $p_1$ and $p_2$ and sample sizes $n_1$ and $n_2$.
• Justify the conditions for the sampling distribution: randomization, the $10\%$ condition (when sampling without replacement), and the large counts condition for approximate normality.
• Describe the sampling distribution of $\hat{p}_1 - \hat{p}_2$ (shape, center, and variability) in context.
• Calculate and interpret probabilities involving $\hat{p}_1 - \hat{p}_2$ using a normal distribution.

Core ideas

Parameters of the sampling distribution

Suppose we take an independent random sample of size $n_1$ from a population with proportion of successes $p_1$, and a second independent random sample of size $n_2$ from a population with proportion of successes (or randomly assign and units to two treatments with true success probabilities and ). The sampling distribution of has:

• Mean (Center):

  $$\mu_{\hat{p}_1 - \hat{p}_2} = p_1 - p_2$$