---
title: "Two proportions: mothers working full-time"
subject: "Probability and Statistics"
chapter: 7
key: "semester_5/probability_and_statistics:day-care-working-mothers-proportion"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/day-care-working-mothers-proportion
site: BCT Engineering
---

# Two proportions: mothers working full-time

Probability and Statistics · Chapter 7: Inference concerning proportion

- Asked in: 2073 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/day-care-working-mothers-proportion
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-7

## Question

Two different areas of a city are being considered as sites for day-care centers. Of 200 households surveys in one section, the proportion in which the mother worked full-time was 0.52. In another section, 40% of 150 households surveyed had mothers at full time jobs. At 0.05 level of significance, is there a significant difference in the proportion of working mothers in the two areas of the city?

## Short answer

For two proportions, pooled p-hat=0.4686, z=(0.52-0.40)/0.0539=2.226. At 0.05 two-tailed critical value is 1.960; since |z| > 1.960, reject H0. There is a significant difference in working-mother proportions.

## Answer

This is a test for the difference of two proportions (large samples, two-tailed).

```text
Area 1: n₁ = 200, p̂₁ = 0.52 → x₁ = 104
Area 2: n₂ = 150, p̂₂ = 0.40 → x₂ = 60
Pooled p̂ = (104 + 60)/350 = 0.4686,  q̂ = 0.5314
```

1. H0: p₁ = p₂ (no difference in the proportion of working mothers)
   H1: p₁ ≠ p₂ (two-tailed)
2. Level of significance: α = 0.05
3. Test statistic:

```text
SE = √( p̂q̂ (1/n₁ + 1/n₂) ) = √(0.4686×0.5314×(0.005 + 0.006667)) = 0.05390
z  = (p̂₁ − p̂₂)/SE = (0.52 − 0.40)/0.05390 = 2.226
```

4. Critical value: z(0.05, two-tail) = ±1.960. Reject H0 if |z| > 1.960.
5. Decision: |z| = 2.226 > 1.960 (p-value = 0.026), so **reject H0**.

**Conclusion:** at the 5% level there is a significant difference in the proportion of households with full-time working mothers in the two areas (higher in the first area).

## Check yourself

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

1. What are x1 and x2?
   A. 52 and 40
   B. 104 and 60
   C. 200 and 150
   D. 0.52 and 0.40
   Answer: B. 0.52 × 200 = 104 and 0.40 × 150 = 60. (Key point: Counts)

2. What is the pooled p̂?
   A. 0.46
   B. (0.52 + 0.40)/2 = 0.46
   C. 164/350 = 0.4686
   D. 0.12
   Answer: C. Pooled p̂ = (104 + 60)/350 = 0.4686, weighted by sample size. (Key point: Pooled proportion)

3. Why two-tailed?
   A. n is large
   B. The proportion is higher in area 1
   C. alpha is 0.05
   D. The question asks for any difference
   Answer: D. 'Significant difference' without direction gives a two-tailed test. (Key point: Hypothesis tail)

4. z = 2.226 vs ±1.960. What is concluded?
   A. Reject H0; the proportions differ significantly
   B. Fail to reject H0
   C. No difference
   D. Inconclusive
   Answer: A. |z| > 1.96, so the proportion is higher in area 1. (Key point: Decision)

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