---
title: "Critical value; water pumps conforming"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:critical-value-water-pumps-95"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/critical-value-water-pumps-95
site: BCT Engineering
---

# Critical value; water pumps conforming

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2075 Asoj, 6 marks; 2073 Shrawan, 5 marks (asked 2 times)
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/critical-value-water-pumps-95
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Define critical value. A manufacturer claimed that at least 95% of the water pumps supplied to the ABC Company conformed to specification. However, the product manager at ABC Company wasn't satisfied with the claim of the manufacturer hence to test the claim, the manager examined a sample of 250 water pumps supplied last month and found that 228 water pumps as per the specification. Can you conclude that the production manager is right to doubt on the claim of the manufactures (α = 0.01)

## Short answer

Test H0: p=0.95 vs H1: p<0.95. Sample p̂=228/250=0.912, SE=√(0.95×0.05/250)=0.013784, z=−2.757. Critical value at 1% is −2.326; reject H0, manager’s doubt is justified.

## Answer

**Critical value:** the value of the test statistic (read from the z, t, χ² or F table for the chosen α and degrees of freedom) that separates the rejection region from the acceptance region. If the calculated statistic falls beyond it, H0 is rejected. For example, ±1.96 for a two-tailed z-test at α = 0.05.

#### Test of the claim

The claim is p ≥ 0.95, so the doubt is that p < 0.95 (left-tailed test, large sample).

```text
n = 250, x = 228, p̂ = 228/250 = 0.912,  p₀ = 0.95, q₀ = 0.05
```

1. H0: p = 0.95 (the claim holds, at least 95% conform)
   H1: p < 0.95 (fewer than 95% conform; the manager's doubt)
2. Level of significance: α = 0.01
3. Test statistic:

```text
SE = √(p₀q₀/n) = √(0.95×0.05/250) = √0.00019 = 0.013784
z  = (p̂ − p₀)/SE = (0.912 − 0.95)/0.013784 = -2.757
```

4. Critical value: z(0.01, left-tail) = −2.326. Reject H0 if z < −2.326.
5. Decision: z = -2.757 < −2.326 (p-value = 0.0029), so **reject H0**.

**Conclusion:** at the 1% level the sample does not support the manufacturer's claim. The production manager is right to doubt that at least 95% of the water pumps conform to the specification.

## Check yourself

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

1. What is a critical value?
   A. The sample statistic
   B. The population mean
   C. The p-value
   D. The value separating rejection and acceptance regions
   Answer: D. It is read from the table for alpha and d.f.; a statistic beyond it rejects H0. (Key point: Critical value)

2. The claim is 'at least 95%' and the manager doubts it. What is H1?
   A. p < 0.95 (left-tailed)
   B. p ≠ 0.95
   C. p = 0.95
   D. p > 0.95
   Answer: A. Doubting an 'at least' claim means believing the proportion is lower. (Key point: Direction of test)

3. Which SE is used in the single-proportion test?
   A. sqrt(p̂(1−p̂)/n)
   B. sqrt(p0q0/n)
   C. p0/n
   D. sqrt(n p0)
   Answer: B. Under H0 the proportion is p0, so SE = sqrt(0.95 × 0.05/250). (Key point: Standard error)

4. The answer gives z = −2.757 vs critical −2.326. What is the decision?
   A. Fail to reject H0
   B. Use two-tailed ±2.576 and accept
   C. Reject H0; the manager is right to doubt the claim
   D. Not enough data
   Answer: C. −2.757 < −2.326, so H0 is rejected. (Key point: Decision)

## Related in Probability and Statistics

- [Confidence level; cables conforming claim](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aconfidence-level-cables-95-claim) — Probability and Statistics, chapter 7
- [Test of proportion: 95% equipment claim](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aequipment-95-percent-conform-test) — Probability and Statistics, chapter 7
- [Test of proportion: pumps needing repairs](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apumps-repairs-30-percent-claim) — Probability and Statistics, chapter 7