- 2075 Chaitra · 5 marks
CI for difference in bulb life of two brands
Short answer
Difference in means = 58 h, SE = 11.11. 95% CI: (36.2, 79.8) hours; 99% CI: (29.4, 86.6) hours. Both intervals exclude zero, suggesting brand A has longer mean life.
A whole sale dealer wanted to buy a large quantity of light bulbs from two brands label A and B. He bought 100 bulbs from each brand and found by testing that brand A had mean life time 1120 hours and standard deviation 75 hours and brand B had mean life time 1062 hours and standard deviation 82 hours. Find the 95% and 99% confidence limits for the difference in the average life of bulbs from the two brands.
Given: Brand A: n₁ = 100, x̄₁ = 1120, s₁ = 75; Brand B: n₂ = 100, x̄₂ = 1062, s₂ = 82. Large samples, so use the normal distribution.
Difference of means: x̄₁ − x̄₂ = 1120 − 1062 = 58 hours
Standard error: SE = √(s₁²/n₁ + s₂²/n₂) = √(75²/100 + 82²/100) = √(56.25 + 67.24) = √123.49 = 11.113
Confidence limits: (x̄₁ − x̄₂) ± Z_α/2 · SE
95%: Z = 1.96. Margin = 1.96 × 11.113 = 21.78 Limits = 58 ± 21.78 = (36.22, 79.78)
99%: Z = 2.576. Margin = 2.576 × 11.113 = 28.63 Limits = 58 ± 28.63 = (29.37, 86.63)
Answer: 95% limits: 36.2 to 79.8 hours; 99% limits: 29.4 to 86.6 hours. Zero lies outside both intervals, so brand A has a significantly longer mean life than brand B.
✓ Answer checked by Claude Sonnet 5.5
Related questions
- Test: two bulb brands differ in quality · Probability and Statistics, chapter 6
- Bulb life of two companies: grouped data · Probability and Statistics, chapter 6 · asked 2×
- Normal: bulb life between 750 and 825 h · Probability and Statistics, chapter 3
- Test: wear of two tyre brands · Probability and Statistics, chapter 6
- Hypothesis testing; difference of two proportions · Probability and Statistics, chapter 7 · asked 2×
- CLT: average life of 16 bulbs · Probability and Statistics, chapter 4 · asked 2×