Track 7 · Estimation and Confidence Intervals · Lesson 4 of 10
Interval for a mean with unknown σ
Build and interpret a t confidence interval for a mean, explain why t replaces z when σ is estimated, and judge when the interval can be trusted.
A clinic trialling a new blood-pressure tablet has 91 patients' readings and no idea of the true standard deviation — it must be estimated from the same data. Using the normal critical value as if σ were known would make the interval too narrow, and the study would claim more precision than it has.
In practice is unknown, so we plug in the sample SD and use the estimated standard error . But is itself a random quantity: in some samples it is too small, in others too large. When it is too small, is too narrow and misses more often than 5% of the time.
The fix, found by W. S. Gosset in 1908, is to use the exact distribution of the standardised mean with in it. For a normal population,
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It has 7 graded exercises with feedback, hints and worked solutions. After it you will be able to:
- Use t with n − 1 degrees of freedom whenever σ is estimated by s
- Build a t interval from raw data or from summary statistics
- Explain why z with s undercovers in small samples
- Judge whether a t interval can be trusted from the sample size, skew and outliers