---
title: "Test: mean weights of two processes"
subject: "Probability and Statistics"
chapter: 6
key: "semester_5/probability_and_statistics:bulbs-factory-two-processes-weights"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/bulbs-factory-two-processes-weights
site: BCT Engineering
---

# Test: mean weights of two processes

Probability and Statistics · Chapter 6: Inference concerning mean

- Asked in: 2071 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-factory-two-processes-weights
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-6

## Question

In a certain factory, there are two independent processes manufacturing the same item. The average weight in a sample of 250 items produced from one process is found to be 120 gram with a standard deviation of 12 gram, while the corresponding figures in a sample of 400 items from the other process are 124 and 14 respectively. Test whether the two mean weights differ significantly or not at 5 percent level of significance.

## Short answer

Two-sample z-test: SE = √(12²/250+14²/400)=1.0325, z = (120−124)/1.0325 = −3.874. Since |z|>1.96, reject H0; the two mean weights differ significantly.

## Answer

Both samples are large, so the two-sample z-test is applied.

1. H0: μ₁ = μ₂ (mean weights are equal)
   H1: μ₁ ≠ μ₂ (two-tailed)
2. Level of significance: α = 0.05
3. Test statistic (large samples, z-test):

```text
z = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)
SE = √(12²/250 + 14²/400) = √(0.5760 + 0.4900) = 1.0325
z  = (120 − 124) / 1.0325 = -3.874
```

4. Critical value: z at α = 0.05 (two-tail) = ±1.960. Reject H0 if |z| > 1.960.
5. Decision: the calculated value falls in the rejection region (p-value = 0.0001), so **reject H0**.

**Conclusion:** the two mean weights differ significantly; the second process gives heavier items.

## Check yourself

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

1. Which test is used?
   A. Paired t-test
   B. One-way ANOVA
   C. Chi-square test
   D. Two-sample z-test because both samples are large
   Answer: D. Independent large samples (250 and 400) use the two-sample z-test. (Key point: Choice of test)

2. What is SE?
   A. sqrt(12 + 14)
   B. sqrt(12^2/250 + 14^2/400) = 1.0325
   C. (12 + 14)/650
   D. sqrt(144 + 196) = 18.4
   Answer: B. SE = sqrt(0.576 + 0.49) = 1.0325. (Key point: Standard error)

3. What does z = -3.874 imply at the 5% level?
   A. Do not reject H0
   B. Test is inconclusive
   C. Reject H0 since |z| > 1.96
   D. Reject H0 only at 1%
   Answer: C. |z| = 3.874 > 1.96 lies in the rejection region. (Key point: Decision)

4. What is the conclusion?
   A. The two mean weights differ significantly; the second process is heavier
   B. The first process is heavier
   C. Weights are equal
   D. Need a larger sample
   Answer: A. H0 is rejected and 124 g > 120 g. (Key point: Conclusion)

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