Justifying a Claim Based on a Confidence Interval for a Population Proportion

Two sentences that mean different things

The interval. "We are 95% confident that the interval from 0.480 to 0.560 contains the true proportion of all the city's voters who support the transit tax." This is about one interval and one parameter, and the parameter is described with its population, in context.

The confidence level. "If we took many random samples of 600 voters and built a 95% confidence interval from each, about 95% of those intervals would capture the true proportion of the city's voters who support the tax." This is about the method, not about the interval you got.

Exam questions ask for these two separately. The second explains the first. Your interval came from one sample, so it either contains the true proportion or it does not, and you will never know which. What you do know is that the method that produced it captures the true proportion about 95% of the time. That track record is what the word confident means.

Plausible values and claims

Every value inside a confidence interval is a plausible value for the parameter: a value the data are consistent with. Values outside it are not plausible at that confidence level. That makes an interval a tool for judging a claim about $p$.

• If a claimed value is outside the interval, the interval gives convincing evidence that the true proportion is different from the claimed value, and it tells you in which direction.
• If a claimed value is inside the interval, the claim is plausible. The data give no convincing evidence against it, but they do not confirm it either, because every other value in the interval is plausible too.
• A claim about a range of values, such as "a majority" or "fewer than 1 in 4", is supported only when the whole interval lies inside that range.

What makes an interval wider or narrower

The margin of error is $z^{\ast }\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$, and two choices you make control its size.

For a given sample, raising the confidence level raises the critical value $z^{\ast }$. That raises the margin of error and widens the interval. More confidence costs precision: a 99% interval captures $p$ more often precisely because it is wider.

Raising the sample size, with everything else the same, lowers the standard error and narrows the interval. The width is roughly proportional to $\frac{1}{\sqrt{n}}$, so cutting the width in half takes four times as many people, not twice as many. In the table, read down a column to see the sample size effect and across a row to see the confidence level effect.