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9. Hypothesis tests

A mean with σ known (z), a mean with s (t), or a proportion. Pick the direction of H₁ and α; get the statistic, the p-value and the decision in words.

Test

H₁

H₀ value, μ₀ or p₀

Sample size n

Sample mean x (mean tests)

σ or s (mean tests)

Count with the trait (proportion)

HypothesesH₀: μ = 250 H₁: μ ≠ 250H₀ holds the equals sign. H₁ is the claim you want evidence for.
Standard error1σ / √n = 6 / √36
Test statistic z-2.5(x bar − 250) / 1
Critical valuez = ±1.960Two-sided: α/2 = 0.025 in each tail. Reject if the statistic is past it.
p-value0.01242 × P(z > |-2.5|)
DecisionReject H₀p-value 0.0124 < α = 0.05
In wordsThe data are strong evidence that μ ≠ 250.Say it about the population, at α = 0.05.
95% interval for μ245.54 to 249.46Leaves out 250, which matches the decision. A two-sided test and its interval always agree.