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8. Confidence intervals and sample size

Mean with s (t), mean with σ known (z), or a proportion. Then the sample size for a target margin.

Type

Confidence

Sample size n

Mean x (mean types)

SD s or σ (mean types)

Count with the trait (proportion)

Target E (for sample size)

E is

Critical valuet = 2.064t with n − 1 = 24 df, 95% two-sided. σ unknown, so t.
Standard error1.24s / √n = 6.2 / √25
95% interval for μ39.9406 to 45.0594x bar ± 2.064 × 1.24 = 42.5 ± 2.5594
Margin of error2.55942.064 × 1.24
Sample size for ± 2 (z)37(zσ/E)² = (1.96 × 6.2 / 2)² = 36.9178. Round up. The usual textbook answer.
Same, if you keep t = 2.06441(2.064 × 6.2 / 2)² = 40.9395. Check which one your book or teacher uses.
What it meansWe are 95% confident the true value is in this rangeNot a 95% chance for this one interval; 95% of intervals built this way catch the truth.