Building a One-Proportion z-Interval

Every confidence interval has the same shape: estimate plus or minus margin of error. That's because a single sample proportion, p-hat, is almost certainly not exactly equal to the true proportion p. Sampling variability means p-hat bounces around p from sample to sample. So instead of reporting one number and pretending it's exact, you report a range of plausible values for p, centered on your best guess, p-hat.

The margin of error itself is built from two pieces multiplied together: the critical value z·and the standard error of p-hat. The standard error, √(p-hat(1-p-hat)/n), measures how much p-hat typically varies from sample to sample. The critical value z·comes from the standard normal curve and controls how wide the interval needs to be to capture the true p with the confidence level you want, 95 percent, 99 percent, or otherwise. A higher confidence level demands a larger z*, which stretches the interval wider, because you're demanding more certainty that you've trapped the true p. Multiply z·by the standard error and you get the margin of error. Add and subtract that from p-hat and you get the interval: p-hat - z*√(p-hat(1-p-hat)/n) to p-hat + z*√(p-hat(1-p-hat)/n).

This interval is the foundation for everything else in the unit. Once you can build one interval for one proportion, the same logic extends to comparing two proportions, and the same critical-value-times-standard-error structure reappears in hypothesis tests, just repackaged as a test statistic instead of a margin of error.