4.3 Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

Template: "We are C% confident that the interval from a to b captures the true mean [variable] for [population]." For the sleep example: "We are 95% confident that the interval from 6.24 to 7.36 hours captures the true mean sleep on school nights for all students at the school."

For matched pairs, name the mean difference and its order: "…captures the true mean improvement (after − before) in practice test score."

The interval comes from a sample, so it may or may not contain μ. That uncertainty is why you say "confident."

"If we took many random samples of 20 students and built a 95% t-interval from each, about 95% of those intervals would capture the true mean sleep for all students at the school." The 95% is a success rate for the method.

Values inside are plausible; values outside aren't. If the school nurse claims students average 8 hours of sleep, the interval (6.24, 7.36) says that's not plausible: 8 is above the whole interval. A claim of 7 hours is plausible.

For paired data, check whether 0 is in the interval. The improvement interval (1.60, 10.15) is entirely above 0, so it gives convincing evidence that the review session raises scores on average. If 0 were inside, "no improvement" would be plausible.

To plan a margin of error, remember that quadrupling the sample size roughly halves the width.

• Higher confidence level: larger t*, larger margin of error, wider interval.
• Larger sample: smaller SE (s/√n) and slightly smaller t* (more df), so a narrower interval. Width is roughly proportional to 1/√n.
• More variable data (larger s): wider interval.

The sleep interval had margin of error 0.56 hours with n = 20. With the same s = 1.2 and n = 80 (four times as many students), SE = 1.2/√80 ≈ 0.134 and t* ≈ 1.990 (df = 79), so the margin of error drops to about 0.27 hours, roughly half. It's a bit less than half because t* also shrinks as df grows.

The interval and a two-sided test agree: a 95% interval contains exactly the values of μ₀ that a two-sided test at α = 0.05 would not reject.

The interval estimates the population mean. It does not say 95% of students sleep between 6.24 and 7.36 hours. Individual students vary far more than that (the sample standard deviation was 1.2 hours).