Carrying Out a Test for a Population Proportion

A test runs in four steps: state the parameter, hypotheses, and $\alpha$; plan by naming the procedure and checking conditions with the study's numbers; compute the test statistic and p-value; then compare, decide, and interpret in context. For $H_0: p = 0.40$ against $H_a: p < 0.40$ with $n = 250$ and 88 renewals, $z \approx -1.55$ and the p-value is about 0.061, so at $\alpha = 0.05$ we fail to reject and report that there is not convincing evidence the renewal rate is below 40%.

Two asymmetries define the topic. A large p-value never becomes support for the null: the data are consistent with 0.40 and equally with 0.38 and 0.36, so accepting $H_0$, proving no difference, or stating that the rate is 40% all convert an absence of evidence into a finding. And statistical significance measures detectability, not size: $\hat{p} = 0.508$ with $n = 40{,}000$ gives a two-sided p-value near 0.0014 for an eight-tenths-of-a-point effect, while $\hat{p} \approx 0.633$ with $n = 30$ gives about 0.072 for a 13-point one. The threshold is a line, not a cliff, so 0.049 and 0.051 describe nearly the same evidence.