Setting Up a Test for a Population Proportion

A one-proportion z-test is set up with hypotheses about the parameter: $H_0: p = p_0$ with equality, against $H_a$ using $<$, $>$, or $\ne$, with the direction taken from the research question before any data are seen. For a company claiming a 40% renewal rate and a manager who suspects it is lower, $H_0: p = 0.40$ against $H_a: p < 0.40$, where $p$ is the renewal proportion among all customers. The conditions are random, $n \le 0.10N$, and $np_0 \ge 10$ with $n(1-p_0) \ge 10$, using the hypothesized proportion because the test reasons inside the world where the null is true. The significance level is chosen in advance.

The hypotheses are the failure point. They get written about $\hat{p}$, a number that will be known exactly and so cannot be hypothesized about, or about the 250 people surveyed rather than the population they came from. The alternative gets its direction from the sample: seeing 0.352 and then choosing $p < 0.40$ puts the whole significance level in a tail the data already selected. The null gets an inequality when it must carry equality. And the conditions get checked with $\hat{p}$, which belongs to intervals, or recited without the study's numbers at all.