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

Using a CI to evaluate a claim

A confidence interval gives a range of plausible values for a population proportion. To justify a claim, check whether the claimed value falls inside or outside the interval. If a value is outside the interval, it is not plausible at that confidence level; if inside, it's a plausible value of p. Always connect back to context.

Capturing the parameter, not the estimate

A confidence interval is a statement about the population proportion (p), not the sample proportion. Interpretation: 'We are C% confident that the interval from ___ to ___ captures the true proportion of context.' This means if we repeated the sampling process many times, about C% of intervals would capture the true parameter.

Two intervals, comparing groups

When comparing two population proportions using a confidence interval for p1 − p2, check whether 0 is contained in the interval. If 0 is not in the interval, this supports a claim that the proportions differ; the sign of the interval tells you the direction of the difference.

Key terms

How do you justify a claim using a CI for p?

Check if the claimed value lies inside or outside the interval; if outside, the claim is not plausible at that confidence level.

Interpreting a confidence interval

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

Using a CI for p1 − p2 to compare groups

If 0 is not in the interval, there's convincing evidence the two proportions differ; the sign shows which group has the higher proportion.

What does a value outside the CI mean?

It is not a plausible value of the population parameter at the stated confidence level.

What does a CI capture — parameter or statistic?

The population parameter (p); the sample proportion is just the point estimate used to build the interval.

Try an AP-style question

A 95% CI for the proportion of voters who support a bill is (0.42, 0.51). Is there convincing evidence that a majority support it?

1. No — values at or below 0.5 are still plausible
2. Yes — the midpoint is close to 0.5
3. Yes — the interval includes values above 0.5
4. No — the interval is too narrow