Does the new process cut defects below 50 per batch?

Question

A production line averages 50 defects per batch with a known standard deviation of $\sigma = 8$ . An engineer claims a new process reduces defects. A sample of 16 batches averages 46 defects.

Run a one-sided z-test at the 5% level (do defects decrease?): report the z-statistic and the decision.

Accepted Answer

Directional claim (defects go down), \sigma known, so use a lower-tail z-test. Hypotheses: H0: \mu = 50 versus H1: \mu < 50. Standard error: 	ext{SE} = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2. Test statistic: z = \frac{46 - 50}{2} = \frac{-4}{2} = -2.0. For a lower-tail test the critical value is -z{0.05} = -1.645. Since -2.0 < -1.645, reject H0. The one-sided p-value is \Phi(-2.0) = 0.0228. Conclusion: there is evidence the new process lowers the mean defect count. Takeaway. For a "reduces" claim the rejection region sits in the lower tail, so you reject when z is sufficiently negative. Alway…

Does the new process cut defects below 50 per batch?

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