Is the pass rate below the promised 95%?

Question

A supplier guarantees that at least 95% of parts pass inspection. A buyer inspects 400 parts and finds 368 pass (92%).

Run a one-sided one-proportion z-test at the 5% level (is the pass rate below 95%?): report the z-statistic and the decision.

Accepted Answer

The buyer's concern is directional (rate below 95%), so use a lower-tail test. Hypotheses: H0: p = 0.95 versus H1: p < 0.95. Sample proportion: \hat p = 368/400 = 0.92. Standard error under the null (p = 0.95): 	ext{SE} = \sqrt{\frac{0.95 	imes 0.05}{400}} = \sqrt{0.00011875} = 0.01090. Test statistic: z = \frac{0.92 - 0.95}{0.01090} = \frac{-0.03}{0.01090} = -2.75. The lower-tail 5% critical value is -1.645. Since -2.75 < -1.645, reject H0. The one-sided p-value is \Phi(-2.75) \approx 0.003. Conclusion: there is evidence the true pass rate falls short of the guaranteed 95%. Takeaway. Even …

Is the pass rate below the promised 95%?

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