Do two factories have different defect rates?

Question

Factory X ships 500 units with 25 defective (5%); factory Y ships 400 units with 32 defective (8%).

Test at the 5% level (two-sided) whether the true defect rates differ, using a pooled two-proportion z-test, and report the z-statistic.

Accepted Answer

Hypotheses: H0: pX = pY versus H1: pX 
eq pY. Sample proportions: \hat pX = 25/500 = 0.050, \hat pY = 32/400 = 0.080. Pooled proportion: \hat p = \frac{25 + 32}{500 + 400} = \frac{57}{900} = 0.0633. Standard error under the null: 	ext{SE} = \sqrt{0.0633 	imes 0.9367 	imes \left(\frac{1}{500} + \frac{1}{400}ight)} = \sqrt{0.059324 	imes 0.0045} = \sqrt{0.000267} = 0.01634. Test statistic: z = \frac{0.080 - 0.050}{0.01634} = \frac{0.030}{0.01634} = 1.84. Since 1.84 = 1.84 < 1.96, we fail to reject H0. The two-sided p-value is about 0.066. Conclusion: the 3-point gap in defect rates is no…

Do two factories have different defect rates?

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