Two-sided z test: worked example
Compute the statistic, compare with the critical value, decide. "Do not reject H₀" does not mean "H₀ is proven".
Worked example
A maker claims μ = 50. A sample of n = 64 gives x̄ = 50.15; σ = 5 is known. Test H₀: μ = 50 two-sided at α = 5% (critical value 1.96).
Worked solution
- 1.z = (x̄ − μ₀) / (σ / √n) = (50.15 − 50) / (5 / 8) = 0.24
- 2.|z| = 0.24 ≤ 1.96 → do not reject H₀
Typical mistake: Reading "do not reject H₀" as "H₀ is proven".
Your turn
Three problems of exactly this type, each time with new numbers. Comma or point; fractions like 1/6 work too.
Task
A maker claims μ = 50. A sample of n = 64 gives x̄ = 49.53; σ = 8 is known. Test H₀: μ = 50 two-sided at α = 5% (critical value 1.96).
Test statistic z (2 decimal places)
Decision
Comma or point; fractions like 1/6 work too.
Formulas for this topic
- 1. State H₀ and H₁, choose α
- 2. Test statistic: z = (x̄ − μ₀) / (σ / √n) or t = (x̄ − μ₀) / (s / √n)
- 3. Compare with the critical value (two-sided |z|, one-sided only in the direction of H₁)
- 4. Decide: reject or do not reject H₀
Typical mistake: Reading "do not reject H₀" as "H₀ is proven".
Frequently asked questions
What is the typical mistake in "Two-sided z test: worked example"?
Reading "do not reject H₀" as "H₀ is proven".
What is a type I error?
Rejecting H₀ although it is true; its probability is at most α.