Z-interval

A z-interval takes its critical value $z^*$ (z-star) from the standard normal curve, so it is usable when a normal curve is a fair model for the sampling distribution of the statistic. On this course that means a proportion: $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, where $\hat{p}$ (p-hat) is the sample proportion. The conditions are a random sample, a population at least 10 times the sample, and at least 10 successes and 10 failures in the data.

Suppose 60 of 100 randomly sampled adults say yes. Then $\hat{p} = 0.60$, the standard error is $\sqrt{0.60(0.40)/100} = 0.048990$, and at 95% confidence $z^* = 1.96$ gives a margin of error of $1.96(0.048990) = 0.096020$. The interval runs 0.504 to 0.696. The counts are 60 successes and 40 failures, so the condition passes.

One error belongs to intervals specifically. The check here uses the observed counts, $n\hat{p} = 60$ and $n(1-\hat{p}) = 40$, because there is no hypothesized proportion anywhere in an interval. Students carry over the test version and write $np_0 \geq 10$ using a claimed value of $p$, which for an interval does not exist. The second slip is the probability reading of the 95%, which attaches to the method and not to the endpoints you already computed; what 95% confidence means works through the wording.

The interval is an approximation, and it undercovers. Its exact long-run coverage, computed from the binomial distribution, is 94.1% at $n = 50$ and $p = 0.60$, and 87.6% at $n = 20$ and $p = 0.90$, against a nominal 95% in both cases. That shortfall is the reason for the counts condition, and the plus-four interval is the standard repair outside AP.

Which parameter takes which interval, and why a mean takes $t^*$ instead, is set out in z-interval vs t-interval. Building this one is Unit 3, topic 3.3, Constructing a Confidence Interval for a Population Proportion.