4.2 Confidence Intervals for the Mean of a Population

Building a confidence interval for μ

Since population σ is usually unknown, we use the t-distribution instead of the normal distribution. A confidence interval for a mean is x̄ ± t\* · (s/√n), where t\* is the critical value from the t-distribution with df = n − 1 and s/√n is the standard error of the mean.

Conditions for inference

Check: Random (data from random sample/assignment), Independence (10% condition if sampling without replacement), and Normality — population is normal, or n ≥ 30 (CLT), or a graph of the sample data shows no strong skew/outliers for smaller n. Meeting these ensures the sampling distribution of x̄ is approximately t-distributed.

Interpreting the interval

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

Key terms

Confidence interval formula for μ

x̄ ± t*(s/√n), where t* has df = n − 1

Why use t instead of z?

Because σ is unknown and estimated by s, adding extra uncertainty that requires a wider, t-based distribution.

Degrees of freedom (one-sample t)

df = n − 1

Conditions for a t-interval

Random, Independent (10% condition), and Normal/Large Sample (population normal, n ≥ 30, or no strong skew/outliers)

Correct interpretation of confidence level

In repeated random sampling with the same method, about C% of resulting intervals would capture the true population mean μ.

Try an AP-style question

Why use t* instead of z* when building a CI for μ?

1. The sample is always small
2. σ is unknown and estimated by s
3. t* is always smaller than z*
4. The data are categorical