---
title: "Test: two bulb brands differ in quality"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:light-bulbs-two-brands-differ-quality"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/light-bulbs-two-brands-differ-quality
site: BCT Engineering
---

# Test: two bulb brands differ in quality

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2068 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/light-bulbs-two-brands-differ-quality
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

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%.

## 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.

## Answer

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:

```text
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
```

4. Critical value: z(0.10, two-tail) = ±1.645. Reject H0 if |z| > 1.645.
5. 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.

## Check yourself

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

1. Why is a two-sample z-test used for the two bulb brands?
   A. The population is Poisson
   B. The samples are small and variances are equal
   C. The data are paired
   D. Both samples are large (n = 50 each)
   Answer: D. With n = 50 in each sample the sampling distribution of the difference of means is approximately normal, so z is used with s in place of sigma. (Key point: Choice of test)

2. The buyer asks whether the brands DIFFER in quality. What is H1?
   A. mu_A ≠ mu_B (two-tailed)
   B. mu_A < mu_B (left-tailed)
   C. mu_A = mu_B
   D. mu_A > mu_B (right-tailed)
   Answer: A. 'Differ' in either direction gives a two-tailed alternative, so the critical value is ±1.645 at alpha = 0.10. (Key point: Alternative hypothesis)

3. Which standard error is used for z here?
   A. (s1 + s2)/sqrt(n1 + n2)
   B. sqrt(s1²/n1 + s2²/n2)
   C. sqrt(s1² + s2²)/n
   D. s1/sqrt(n1)
   Answer: B. The variances of the two sample means add: SE = sqrt(80²/50 + 94²/50) = 17.456. (Key point: Test statistic)

4. z = 4.239 and the critical value is ±1.645. What is the decision?
   A. Fail to reject H0; the brands do not differ
   B. The test is inconclusive
   C. Reject H0; the brands differ significantly
   D. Accept H1 only if alpha = 0.01
   Answer: C. |z| = 4.239 > 1.645 lies in the rejection region, so H0 is rejected and the quality differs. (Key point: Decision)

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