Carrying Out a One-Proportion z-Test

The test statistic uses p0, the null proportion, instead of the sample proportion phat, because the whole point is to ask how surprising phat would be if the null claim were true. The standard error measures spread under an assumption, and the assumption on trial is p0, not whatever you happened to observe. Build the standard error from phat and you would be measuring variability under a world you have no reason to believe in.

Every test for a proportion follows the same four steps. First, state hypotheses: H0: p = p0 against Ha: p < p0, p > p0, or p does not equal p0. Second, check conditions: the sample must be random, the population at least 10 times the sample size, and both n times p0 and n times (1 minus p0) at least 10, so the sampling distribution of phat is close to normal. Third, calculate: find z = (phat - p0) / √(p0(1-p0)/n), then convert z to a p-value using the normal curve. Fourth, conclude in context: compare the p-value to your significance level and state whether the data give convincing evidence against H0, always tying the answer back to the original claim.

This sits right after building confidence intervals for one proportion, and the two share the same standard error formula with one key difference: intervals use phat in the standard error because there is no claimed value to test, while tests use p0 because there is. Next comes comparing two proportions, where the same four-step logic reappears with a pooled proportion replacing p0, since under the null the two groups are assumed to share one common proportion.