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Test of two proportions: Dyson suppliers

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.

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.

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).

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:
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
  1. Critical value: z(0.05, left-tail) = −1.645. Reject H0 if z < −1.645.
  2. 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.

✓ Answer checked by Claude Sonnet 5.5

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