---
title: "Test of two proportions: Dyson suppliers"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:dyson-hair-product-defective-test"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/dyson-hair-product-defective-test
site: BCT Engineering
---

# Test of two proportions: Dyson suppliers

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2076 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/dyson-hair-product-defective-test
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Dyson Company in Berlin plans to produce a new hair product known as Dyson Supersonic. The suppliers for the company A and Company B found that 1% and 2% defective of 200 and 300 items respectively. Arrange appropriate hypothesis testing to investigate whether Company B is better by using a 0.05 level of significance.

## Short answer

Test H0: pA=pB vs H1: pB<pA. Pooled p̂=8/500=0.016, SE=0.01145, z=(0.02−0.01)/0.01145=0.873. Critical left-tail z=−1.645; fail to reject H0, no evidence Company B is better.

## Answer

Company B is "better" if its proportion of defective items is lower than A's. This is a one-tailed test for the difference of two proportions (large samples).

```text
Company A: n₁ = 200, p̂₁ = 0.01 → x₁ = 2 defective
Company B: n₂ = 300, p̂₂ = 0.02 → x₂ = 6 defective
```

1. H0: p_A = p_B (both have the same defective rate)
   H1: p_B < p_A (company B has a lower defective rate, so B is better)
2. Level of significance: α = 0.05
3. Test statistic:

```text
Pooled p̂ = (2 + 6)/(200 + 300) = 0.016,  q̂ = 0.984
SE = √( p̂q̂ (1/200 + 1/300) ) = √(0.016×0.984×0.008333) = 0.01145
z  = (p̂_B − p̂_A)/SE = (0.02 − 0.01)/0.01145 = 0.873
```

4. Critical value: z(0.05, left-tail) = −1.645. Reject H0 if z < −1.645.
5. Decision: z = +0.873 is not less than −1.645 (p-value = 0.81), so it lies in the acceptance region: **fail to reject H0**.

**Conclusion:** there is no evidence that company B is better. In fact B's sample defective rate (2%) is higher than A's (1%), though the difference is not significant (a two-tailed test also gives |z| = 0.873 < 1.96). Company B is not shown to be better.

## Check yourself

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

1. 'Company B is better' means lower defect rate. What is H1?
   A. p_B > p_A
   B. p_B ≠ p_A
   C. p_B = p_A
   D. p_B < p_A (left-tailed)
   Answer: D. Better means fewer defectives, so H1: p_B < p_A, a one-tailed test. (Key point: Direction of test)

2. How are the numbers of defectives obtained?
   A. x = p̂ × n: 2 for A and 6 for B
   B. x = n/p̂
   C. x = p̂ + n
   D. x = 1 and 2
   Answer: A. 0.01 × 200 = 2 and 0.02 × 300 = 6, giving the pooled p̂ = 8/500 = 0.016. (Key point: Defective counts)

3. What is the critical value for the left-tailed test at 0.05?
   A. −1.96
   B. −1.645
   C. +1.645
   D. −2.326
   Answer: B. A one-tailed 5% left test uses −1.645. (Key point: Critical value)

4. z = +0.873. What is the conclusion?
   A. B is significantly better
   B. Reject H0
   C. Fail to reject H0; B is not shown to be better
   D. A is significantly better
   Answer: C. z is positive and not below −1.645; B's sample rate is actually higher. (Key point: Decision)

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