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Test: two bulb brands differ in quality

Short answer

Two-sample z-test: z = (1282−1208)/√(80²/50+94²/50) = 4.24. Critical z(0.10 two-tailed)=±1.645. Since 4.24 > 1.645, reject H0; brands differ significantly in mean life.

A potential buyer of light bulbs bought 50 bulbs each of two brands. Upon testing these bulbs, he found that brand A had a mean life of 1282 hours with S.D. of 80 hours whereas the brand B had a mean life of 1208 hours with S.D. of 94 hours. Can the buyer be quite certain that the two brands do differ in quality? α = 10%.

Both samples are large (50 each), so a two-sample z-test for the difference of means is used.

  1. H0: μ_A = μ_B (the two brands do not differ in quality) H1: μ_A ≠ μ_B (two-tailed)
  2. Level of significance: α = 0.10
  3. Test statistic:
z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
SE = √(80²/50 + 94²/50) = √(128.00 + 176.72) = 17.4562
z  = (1282 − 1208)/17.4562 = 4.239
  1. Critical value: z(0.10, two-tail) = ±1.645. Reject H0 if |z| > 1.645.
  2. Decision: calculated |z| = 4.239 > 1.645 (p-value < 0.0001), so reject H0.

Conclusion: at the 10% level the two brands differ significantly in mean life. Brand A (1282 h) is better than brand B (1208 h), so the buyer can be confident that the quality differs.

✓ Answer checked by Claude Sonnet 5.5

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