Estimation, Confidence Intervals and Tests

• Unbiased estimates: x̄ for μ; s² = (Σx² − nx̄²)/(n − 1) for σ².
• Standard error of the mean: σ/√n.
• Central limit theorem: for large n, X̄ ≈ N(μ, σ²/n) whatever the population.
• Confidence interval for μ: x̄ ± z σ/√n.
• Difference of means: z = (x̄₁ − x̄₂)/√(σ₁²/n₁ + σ₂²/n₂).

Dividing by n − 1

Use n − 1 in the variance estimate; the standard error shrinks as the sample grows.

Why the normal appears

Quote the CLT when the population is not known to be normal and n is large.

x̄ ± z σ/√n

95% uses z = 1.96; 90% uses 1.6449; 99% uses 2.5758.

z-tests

State H₀ and H₁ about μ (or μ₁ − μ₂), compute z, compare with the critical value, and conclude in context.