AP Stats 3.4: Justifying Claims from a CI

What topic 3.4 covers

Topic 3.4 is about reading and using the interval you built in topic 3.3, not computing a new one. You interpret the interval in context, interpret the confidence level correctly, use the interval to judge a claim about the population proportion, and describe how sample size, confidence level, width, and margin of error relate. Getting the wording exactly right matters, because a common exam mistake is to describe a probability about one interval instead of the long-run behavior of the method.

Interpreting the interval and the level

For a $C\%$ interval $(a, b)$, the standard interpretation is: you are $C\%$ confident that the interval from $a$ to $b$ captures the true proportion of the response variable in the population. The confidence level describes the method, not one interval: in repeated random sampling with the same sample size, about $C\%$ of the intervals built this way would capture the true proportion. Any single computed interval either does or does not contain $p$; you never know which. Reference the parameter and the population in context so the interpretation is complete.

Justifying a claim

A confidence interval gives a range of plausible values for $p$, which you can use as evidence about a claim. If a claimed value falls outside the interval, the interval provides evidence against that value. If it falls inside, the value stays plausible and the data do not rule it out. For a majority claim, check whether the entire interval sits above 0.50, and for a "no better than chance" claim about a coin, check the position of 0.50 the same way.

Sample size, level, and width

For a fixed sample, raising the confidence level raises the critical value $z^{*}$, which raises the margin of error and widens the interval. Holding other things fixed, increasing the sample size lowers the standard error, so the interval narrows. The width is roughly proportional to $\dfrac{1}{\sqrt{n}}$, so cutting the width in half takes about four times the sample size.

There is a trade-off: a higher confidence level buys more certainty that the method captures $p$ but pays for it with a wider, less precise interval. In practice, researchers pick the confidence level first, based on how costly a miss would be, and then choose a sample size large enough to keep the interval narrow enough to be useful.