---
title: "Test of proportion: pumps needing repairs"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:pumps-repairs-30-percent-claim"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pumps-repairs-30-percent-claim
site: BCT Engineering
---

# Test of proportion: pumps needing repairs

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2070 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/pumps-repairs-30-percent-claim
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

A manufacturer of submersible pumps claims that at most 30% of the pumps require within the first 5 years of operation. If a random sample of 120 of the pumps includes 47 which required repairs within the first 5 years, test the null hypothesis p = 0.30 against the alternative hypothesis p > 0.30 at the 0.05 level of significance.

## Short answer

Use right-tailed z-test for a proportion. p̂ = 47/120 = 0.3917, p₀ = 0.30, SE = 0.04183, z = 2.191. Critical z₀.₀₅ = 1.645. Reject H₀: p=0.30. Evidence supports p>0.30, so the manufacturer’s claim is not supported.

## Answer

Large-sample, right-tailed test of a single proportion.

```text
n = 120, x = 47, p̂ = 47/120 = 0.3917,  p₀ = 0.30, q₀ = 0.70
```

1. H0: p = 0.30
   H1: p > 0.30 (more than 30% of pumps need repair)
2. Level of significance: α = 0.05
3. Test statistic:

```text
SE = √(p₀q₀/n) = √(0.30×0.70/120) = 0.04183
z  = (p̂ − p₀)/SE = (0.3917 − 0.30)/0.04183 = 2.191
```

4. Critical value: z(0.05, right-tail) = 1.645. Reject H0 if z > 1.645.
5. Decision: z = 2.191 > 1.645 (p-value = 0.014), so **reject H0**.

**Conclusion:** at the 5% level the data show that more than 30% of the pumps need repair within the first 5 years, so the manufacturer's claim is not supported.

## Check yourself

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

1. Which hypotheses are tested?
   A. H0: p = 0.30, H1: p < 0.30
   B. H0: p = 0.30, H1: p > 0.30
   C. H0: p = 0.30, H1: p not equal 0.30
   D. H0: p > 0.30, H1: p = 0.30
   Answer: B. The question tests p = 0.30 against p > 0.30, a right-tailed test. (Key point: Hypotheses)

2. Which standard error is used in the test statistic?
   A. sqrt(p-hat q-hat / n) with p-hat = 0.3917
   B. sqrt(p0 q0 / n) = 0.04183
   C. p0 / n
   D. sqrt(n p0 q0) = 5.8
   Answer: B. In a test the hypothesised p0 is used: sqrt(0.30 x 0.70 / 120) = 0.04183. (Key point: Standard error under H0)

3. With z = 2.191 and critical z = 1.645 (right tail), what is decided?
   A. Fail to reject H0; the claim stands
   B. Reject H0; more than 30% of pumps need repair
   C. Reject H0; fewer than 30% need repair
   D. Test is two-tailed so the critical value is 1.96, hence fail to reject
   Answer: B. 2.191 > 1.645, so H0 is rejected and the manufacturer's claim is not supported. (Key point: Decision)

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