---
title: "CI for difference in bulb life of two brands"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:bulbs-two-brands-ci-difference"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bulbs-two-brands-ci-difference
site: BCT Engineering
---

# CI for difference in bulb life of two brands

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2075 Chaitra, 5 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bulbs-two-brands-ci-difference
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

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.

## 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.

## Answer

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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Estimate of difference between means?
   A. 58 hours
   B. 1120 hours
   C. 123.49 hours
   D. 11.1 hours
   Answer: A. 1120 − 1062 = 58. (Key point: Point estimate)

2. SE of the difference?
   A. √(75²/100 + 82²/100) = 11.11
   B. (75 + 82)/100
   C. √(75² + 82²)
   D. 75/√100
   Answer: A. √(56.25 + 67.24) = 11.113. (Key point: Standard error)

3. 95% limits?
   A. (29.37, 86.63)
   B. (36.22, 79.78)
   C. (46.9, 69.1)
   D. (58, 79.78)
   Answer: B. 58 ± 1.96(11.113). (Key point: 95% limits)

4. Why is brand A significantly longer-lived?
   A. Zero lies outside both intervals
   B. Zero lies inside
   C. Intervals equal
   D. SE is zero
   Answer: A. Both intervals exclude 0. (Key point: Interpretation)

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