Module 5 · Inference · Topic 6

Sample Size Determination

Key formulas

n = (z* × σ / ME)²

n = (z* / ME)² × p̂(1 − p̂). If p is unknown, use p̂ = 0.5 for the most conservative (largest) n.

Confidence Interval for a Mean: xbar +/- critical value * standard error — Estimate plus or minus a margin of error.
One-Sample Z Test: z = (xbar - mu_0) / (sigma / sqrt(n)) — Tests a sample mean against a null mean when population sigma is known.
Power: Power = 1 - beta — Probability of correctly rejecting a false null hypothesis.

Desired ME = 2, σ = 10, 95% confidence (z* = 1.96). n = (1.96 × 10 / 2)² = (9.8)² = 96.04. Round up to n = 97.

Always round the sample size up to the next whole number. Rounding down would give a margin of error larger than desired.

P-value: The probability of results this extreme or more extreme, assuming the null hypothesis is true. Example: A p-value of 0.03 is evidence against H0 at alpha = 0.05.
Confidence Level: The long-run success rate of a confidence interval procedure. Example: A 95% method captures the true parameter in about 95% of repeated samples.
Type I Error: Rejecting the null hypothesis when it is actually true. Example: The significance level alpha is the Type I error rate.
Type II Error: Failing to reject the null hypothesis when the alternative hypothesis is true. Example: A false negative in a hypothesis test is a Type II error.
Power: The probability of rejecting the null hypothesis when it is false. Example: Power equals 1 - beta, where beta is the Type II error probability.

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