Intervals and tests for one proportion

A confidence interval for a population proportion p is $\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. The part after $\pm$ is the margin of error: the critical value $z^*$ times the standard error. Common critical values are 1.645 for 90 percent, 1.96 for 95 percent and 2.576 for 99 percent.

Conditions. Random: the data come from a random sample (or a randomized experiment). 10 percent: when sampling without replacement, n is less than 10 percent of the population. Large counts: $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$, so the sampling distribution is approximately normal.

A wider interval comes from a higher confidence level or a smaller sample. To plan a sample size for a margin of error ME, solve $z^*\sqrt{\frac{p^*(1-p^*)}{n}} \leq ME$ for n, using $p^* = 0.5$ when you have no prior estimate, and round up.

A significance test for p starts from $H_0: p = p_0$ and an alternative $H_a$ of $p < p_0$, $p > p_0$ or $p \neq p_0$, chosen before seeing the data. The test statistic is $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, and the large counts check uses $np_0$ and $n(1-p_0)$, because the test assumes $H_0$ is true.

The P-value is the probability, if $H_0$ were true, of a sample proportion at least as extreme as the one observed, in the direction of $H_a$. A two-sided alternative counts both tails. If the P-value is less than the significance level $\alpha$, reject $H_0$; otherwise, fail to reject it.

To interpret an interval, name the confidence level, the endpoints and the parameter in context: "We are 95 percent confident that the interval from 0.342 to 0.438 captures the proportion of all registered voters in the city who support the bond."

To finish a test, compare the P-value with $\alpha$ and then say what that means for the parameter in context. Reject: "there is convincing evidence that" the alternative is true. Fail to reject: "there is not convincing evidence that" the alternative is true. Never say the data prove $H_0$.