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Two proportions: mothers working full-time

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.

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?

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

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:
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
  1. Critical value: z(0.05, two-tail) = ±1.960. Reject H0 if |z| > 1.960.
  2. 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).

✓ Answer checked by Claude Sonnet 5.5

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