3.4 Constructing a Confidence Interval for a Population Proportion

The formula for a proportion CI

A confidence interval for p has the form: statistic ± margin of error, or $\hat{p} \pm z^{*}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$. Here $\hat{p}$ is the sample proportion, $z^{*}$ is the critical value from the standard normal curve matching the confidence level, and the square-root term is the standard error.

Checking conditions first

Before building the interval, verify: Random (random sample/assignment), 10% condition (sample size ≤ 10% of population, for independence), and Large Counts condition ($n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$). These ensure the sampling distribution of $\hat{p}$ is approximately normal, justifying use of $z^{*}$.

The four-step process

AP free-response answers should follow: State (identify parameter and confidence level), Plan (name the procedure, check conditions), Do (calculate the interval), Conclude (interpret in context: 'We are C% confident that the interval from ___ to ___ captures the true proportion of ___').

Key terms

Confidence interval formula for p

$\hat{p} \pm z^{*}\sqrt{\hat{p}(1-\hat{p})/n}$ — sample proportion plus or minus margin of error.

Large Counts condition

Requires $n\hat{p} \geq 10$ and $n(1-\hat{p}) \geq 10$ so the sampling distribution of $\hat{p}$ is approximately normal.

10% condition

Sample size should be no more than 10% of the population size to ensure approximate independence when sampling without replacement.

Four-step process

State, Plan, Do, Conclude — the structure for answering confidence interval free-response questions.

Correct CI interpretation

"We are C% confident that the interval from ___ to ___ captures the true population proportion of ___."

Try an AP-style question

p̂ = 0.6 from n = 100. What is an approximate 95% confidence interval for p?

1. 0.6 ± 0.196
2. 0.6 ± 0.096
3. 0.6 ± 0.024
4. 0.6 ± 0.049