p-values and carrying out a test for a population proportion

What the exam asks

Across both sections, you will be asked to:

• Calculate the standardized test statistic $z$ and the corresponding $p$-value using the standard normal distribution.
• Estimate or calculate a $p$-value from a dotplot or table of a simulated null distribution.
• Interpret a $p$-value in context, explicitly conditioning on the null hypothesis being true.
• Make a formal decision by comparing the $p$-value to a significance level $\alpha$, and justify a conclusion in context about the alternative hypothesis $H_a$.

Core ideas

The null distribution and test statistic

When the conditions for inference are met (randomization, $10\%$ condition when sampling without replacement, and large counts $np_0 \ge 10$ and $n(1-p_0) \ge 10$), the sampling distribution of assuming is true is approximately normal with mean and standard deviation .

The standardized test statistic measures how many standard deviations the observed sample proportion $\hat{p}$ lies from the hypothesized value $p_0$:

Finding the $p$-value

The $p$-value is the probability, calculated assuming $H_0$ is true, of obtaining a sample statistic as extreme as or more extreme than the one observed, in the direction specified by $H_a$.