For H₀：μ=μ₀, calculate the test statistic from the sample mean X̄:
Z = (X̄ − μ₀)/(σ/√n)
If |Z| falls in the rejection region, reject H₀.
Example ① With σ=10, n=25, and X̄=55, test H₀: μ=50 against H₁: μ≠50 at 5%.
Solution:Z = (55−50)/(10/5) = 5/2 = 2.5
Since |2.5| > 1.96, reject H₀. We conclude μ ≠ 50.
Solution:Z = (55−50)/(10/5) = 5/2 = 2.5
Since |2.5| > 1.96, reject H₀. We conclude μ ≠ 50.
Key Point
- The larger the Z-score, the farther it is from H₀
Practice Problems
- For σ=4, n=64, X̄=22, and μ₀=20, find the Z-score for a 5% two-tailed test.
Solutions and Explanations
- Solution:Z=4 (4/0.5=8; check: σ/√n=0.5) → reject H₀