Confidence intervals
A confidence interval (CI) is a range of plausible values for a population parameter based on sample data. A 95% CI means: if you repeated the sampling process many times, 95% of the CIs constructed that way would contain the true parameter. It does not mean “there is a 95% probability the true value is in this interval”—the true value is fixed; only the interval is random.
Two cases
Case 1 — population σ known: use the z-distribution.
Case 2 — population σ unknown: use the t-distribution with df = n − 1.
Worked example 1 — z (known σ)
n = 25, x̄ = 48.2 g, σ = 4 g (known). Compute a 95% CI.
Worked example 2 — t (unknown σ)
n = 10, x̄ = 24.5, s = 3.2. Compute a 95% CI. (t*(9 df, 0.025) = 2.262)
Practice problem
n = 16, x̄ = 100, s = 8. 90% CI. t*(15 df, 0.05) = 1.753.
Show answer
SE = 8/√16 = 2.0. CI = 100 ± 1.753 × 2.0 = 100 ± 3.506 ⇒ (96.49, 103.51).
Common exam traps
- Wider CI ≠ better. Wider CIs reflect more uncertainty (smaller n or larger variability). A narrower CI is more informative, all else equal.
- Use t when σ is unknown, regardless of n. A common mistake: using z for large samples even though σ is estimated from the data. Always use t when s is computed from the sample.
- The CI is for the mean, not for individual values. A 95% CI for the mean is much narrower than a 95% prediction interval for a single future observation.
- Increasing n by 4 halves the margin of error. Since SE = σ/√n, you must quadruple n to cut the CI width in half.
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