Testing a Proportion

A test for a proportion is simply a check of whether a percentage you measured, like 60 heads in 100 flips, is far enough from a claimed percentage to doubt the claim. The formula looks tall, but it's the same "how many standard errors off?" idea: on top, $\hat p - p_0$ (what you saw minus what was claimed), and on the bottom, $\sqrt{\frac{p_0(1 - p_0)}{n}}$ (how much sample proportions normally wobble, which you met in the CLT unit). Same structure as the z-test for a mean, with a different standard error.

Why it works. Recall that a sample proportion is just an average of 0s and 1s, so the central limit theorem makes it roughly normal around the true $p$ with standard error $\sqrt{\frac{p(1 - p)}{n}}$. The test plugs in the claimed value and asks how far out your $\hat p$ landed.

Why $p_0$ in the standard error? The whole test is built on the assumption that $H_0$ is true, and if it is, the true proportion is $p_0$. So the spread of $\hat p$ is computed from $p_0$, not from your data.

Conditions: $np_0 \ge 10$ and $n(1 - p_0) \ge 10$, the same normal-approximation check as before, with $p_0$ in place of $p$.

A common mistake. It's tempting to use $\hat p$ in the standard error, since that's what confidence intervals do. In a test, use $p_0$. Intervals don't assume any value for $p$, so they use the best guess, $\hat p$; tests assume $H_0$, so they use $p_0$.

Testing whether a vaccine's efficacy beats a threshold, whether a new design raises the click-through rate, or whether a defect rate exceeds a contract limit.