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Test: performance of college A vs B

Short answer

Large-sample z-test: SE = √(9²/50+7²/60)=1.561, z = (75−79)/1.561 = −2.562. Critical values ±1.96 at 5% level. Reject H0; there is a significant difference in performance between colleges.

An examination was given to 50 students at college A and 60 students at college B. At A mean grade was 75 with standard deviation of 9. At B mean grade was 79 with a standard deviation of 7. Is there significant difference between the performance of students at A and those at B, given that α = 0.05?

Samples are large (n₁ = 50, n₂ = 60), so a two-sample z-test is used with sample SDs as estimates.

  1. H0: μ_A = μ_B (no difference in performance) H1: μ_A ≠ μ_B (two-tailed)
  2. Level of significance: α = 0.05
  3. Test statistic (large samples, z-test):
z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
SE = √(9²/50 + 7²/60) = √(1.6200 + 0.8167) = 1.5610
z  = (75 − 79) / 1.5610 = -2.562
  1. Critical value: z at α = 0.05 (two-tail) = ±1.960. Reject H0 if |z| > 1.960.
  2. Decision: the calculated value falls in the rejection region (p-value = 0.0104), so reject H0.

Conclusion: there is a significant difference between the performance of students at A and B; college B students performed better.

✓ Answer checked by Claude Sonnet 5.5

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