Does the defect rate exceed the claimed 2%?

Question

A manufacturer claims its defect rate is below 2%. An auditor tests whether the true rate exceeds 2%. In a sample of 1000 items, 28 are defective (2.8%).

Run a one-sided one-proportion z-test at the 5% level (is the defect rate above 2%?): report the z-statistic and the decision.

Accepted Answer

The auditor's concern is directional (rate above 2%), so use an upper-tail test. Hypotheses: H0: p = 0.02 versus H1: p 0.02. Sample proportion: \hat p = 28/1000 = 0.028. Standard error under the null (p = 0.02): 	ext{SE} = \sqrt{\frac{0.02 	imes 0.98}{1000}} = \sqrt{0.0000196} = 0.004427. Test statistic: z = \frac{0.028 - 0.020}{0.004427} = \frac{0.008}{0.004427} = 1.81. The one-sided 5% critical value is 1.645. Since 1.81 1.645, reject H0. The one-sided p-value is 1 - \Phi(1.81) \approx 0.035. Conclusion: there is evidence the true defect rate exceeds 2%. Takeaway. A large sample (1000) is…

Does the defect rate exceed the claimed 2%?

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