4.3 Justifying a Claim Based on a Confidence Interval for a Mean

Interpreting the interval

A confidence interval gives a range of plausible values for the population mean μ. To interpret it: 'We are C% confident that the interval from ___ to ___ captures the true mean [context].' The confidence level describes the long-run capture rate of the method, not the probability that this specific interval is correct.

Using the interval to justify a claim

To evaluate a claim about μ, check whether the claimed value falls inside or outside the confidence interval. If a value is outside the interval, it's implausible; if inside, it's a plausible value of μ. This lets you make decisions like 'the interval suggests the mean is greater than 50' by checking if 50 is below the entire interval.

Duality with significance tests

A confidence interval and a two-sided significance test are linked: if a hypothesized value μ₀ falls outside a C% confidence interval, a two-sided test at significance level α = 1 − C would reject H₀. If μ₀ is inside the interval, you would fail to reject H₀. This connection lets you use one interval to answer many hypothesis questions.

Key terms

How do you interpret a confidence interval for μ?

"We are C% confident that the interval from [lower] to [upper] captures the true population mean [in context]."

What does the confidence level actually measure?

The long-run proportion of intervals, from repeated random samples, that would capture the true parameter — not the probability for one specific interval.

How can a confidence interval be used to justify a claim?

Check whether the claimed value lies inside (plausible) or outside (implausible) the interval.

Duality between CI and significance test

If μ₀ is outside a C% confidence interval, a two-sided test at α = 1 − C rejects H₀; if μ₀ is inside, we fail to reject H₀.

Plausible value

A value for the population parameter that falls within the confidence interval, meaning it is not contradicted by the sample data.

Try an AP-style question

A 95% CI for mean body temperature is (98.1, 98.5) °F. What about the claim μ = 98.6?

1. It is plausible since 98.6 is close to 98.5
2. The interval can't address the claim
3. It is not plausible — 98.6 is outside the interval
4. It is proven true by the interval