Topic 6.3 Justifying a Claim Based on a Confidence Interval for a Population Proportion: use a confidence interval for a proportion to evaluate whether a claimed value is plausible, and discuss the effect of confidence level and sample size on the interval.

What this topic is asking

The College Board (Topic 6.3) wants you to use a confidence interval for a proportion to justify a claim: decide whether a claimed value of $p$ is plausible (is it inside the interval?), and explain how the confidence level and sample size affect the interval's width and the conclusion.

The plausible-values logic

This is the bridge from estimation to decision. You do not need a separate test to assess a single claimed value: check whether it falls within the interval. A two-sided significance test at significance level $\alpha = 1 - C$ reaches the same verdict, which is why Topic 6.3 previews the duality with hypothesis testing in Topic 6.6.

Reading the whole interval, not an endpoint

A common error is to judge a claim by whether it is "close to" an endpoint. The rule is strictly inside or outside. Equally, when a claim is directional ("at least $0.30$" or "more than half"), you must read the entire interval against it. If a claim is "more than $0.50$" and the interval is $(0.52, 0.60)$, every plausible value exceeds $0.50$, so the data support the claim. If the interval is $(0.48, 0.56)$, some plausible values are below $0.50$, so the data do not establish that the proportion exceeds $0.50$, even though the point estimate does. The interval, not the point estimate, carries the justification.

How confidence level changes the interval

Raising confidence from $90\%$ to $99\%$ raises $z^{*}$ from $1.645$ to $2.576$, widening the interval around the same $\hat{p}$. The trade-off is real: more confidence (more often correct) costs precision (a wider, less informative interval). A wider interval makes more claimed values plausible, so it is less able to rule a claim out; a narrower one is more decisive but correct less often. Exam questions ask you to predict and explain this direction, not just compute.

How sample size changes the interval

A larger sample shrinks the standard error $\sqrt{\hat{p}(1-\hat{p})/n}$, so for a fixed confidence level the margin of error and width decrease. Precision improves with $\sqrt{n}$: to halve the margin of error you must quadruple the sample. A larger sample therefore yields a narrower interval that can exclude (rule out) more claimed values, sharpening any justification. Crucially, a larger sample changes precision, not the center on average, so it does not bias the estimate toward any claim; it just makes the interval more informative.

Try this

Q1. A $95\%$ interval for $p$ is $(0.22, 0.30)$. Is a claimed value of $p = 0.25$ plausible? [1 point]

• Cue. Yes; $0.25$ lies inside the interval, so it is a plausible value of $p$.

Q2. Holding confidence fixed, how does increasing the sample size affect the ability to rule out a claim? [1 point]

• Cue. It narrows the interval, so more claimed values fall outside and can be ruled out; the justification becomes more decisive.