---
title: "Test of proportion: 95% equipment claim"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:equipment-95-percent-conform-test"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/equipment-95-percent-conform-test
site: BCT Engineering
---

# Test of proportion: 95% equipment claim

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2068 Jestha (old course), 6 marks
- Video: [Hypothesis Testing - Solving Problems With Proportions](https://www.youtube.com/watch?v=76VruarGn2Q) — The Organic Chemistry Tutor
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/equipment-95-percent-conform-test
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

A manufacturer claimed that at least 95% of the equipment, which he supplied to a factory, conformed to specifications. An examination of a sample of 200 pieces of equipment found that 18 are faulty. Test his claim at 5% level of significance.

## Short answer

Using a left-tailed one-proportion z test with p̂=0.91 and p0=0.95, z=-2.596 is below the critical value -1.645; reject H0. The sample does not support the claim that at least 95% conform.

## Answer

Left-tailed test of a single proportion (large sample). Equipment conforming = 200 − 18 = 182.

```text
n = 200, x = 182, p̂ = 182/200 = 0.91,  p₀ = 0.95, q₀ = 0.05
```

1. H0: p = 0.95 (at least 95% conform; claim is true)
   H1: p < 0.95 (fewer than 95% conform; claim is false)
2. Level of significance: α = 0.05
3. Test statistic:

```text
SE = √(p₀q₀/n) = √(0.95×0.05/200) = 0.01541
z  = (p̂ − p₀)/SE = (0.91 − 0.95)/0.01541 = -2.596
```

4. Critical value: z(0.05, left-tail) = −1.645. Reject H0 if z < −1.645.
5. Decision: z = -2.596 < −1.645 (p-value = 0.0047), so **reject H0**.

**Conclusion:** the sample does not support the claim; at the 5% level fewer than 95% of the equipment conform to specifications.

## Check yourself

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

1. The claim is that at least 95% conform. Which hypotheses are used?
   A. H0: p = 0.95, H1: p > 0.95
   B. H0: p = 0.95, H1: p < 0.95
   C. H0: p = 0.95, H1: p not equal 0.95
   D. H0: p = 0.91, H1: p < 0.91
   Answer: B. The claim is only violated if p is below 0.95, so the test is left-tailed. (Key point: Setting up the hypotheses)

2. What sample proportion is used?
   A. 18/200 = 0.09 faulty used as p-hat for conforming
   B. 182/200 = 0.91 conforming
   C. 0.95
   D. 200/182
   Answer: B. The claim is about conforming equipment, so x = 200 - 18 = 182 and p-hat = 0.91. (Key point: Sample proportion)

3. z = -2.596 with critical value -1.645. What is the decision?
   A. Fail to reject H0; claim supported
   B. Reject H0; fewer than 95% conform, claim not supported
   C. Reject H0; more than 95% conform
   D. Use the two-tailed value 1.96 so fail to reject
   Answer: B. -2.596 < -1.645 falls in the left rejection region, so the claim is rejected. (Key point: Decision)

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