Does the new alloy exceed 400 MPa tensile strength?

Question

A standard alloy has a mean tensile strength of 400 MPa with a known standard deviation of $\sigma = 30$ . A metallurgist claims a new alloy is stronger. A sample of 36 specimens averages 413 MPa.

Run a one-sided z-test at the 5% level (is it stronger?): report the z-statistic and the decision.

Accepted Answer

Directional claim (stronger), \sigma known, so use an upper-tail z-test. Hypotheses: H0: \mu = 400 versus H1: \mu 400. Standard error: 	ext{SE} = \frac{30}{\sqrt{36}} = \frac{30}{6} = 5. Test statistic: z = \frac{413 - 400}{5} = \frac{13}{5} = 2.6. The one-sided 5% critical value is z{0.05} = 1.645. Since 2.6 1.645, reject H0. The one-sided p-value is 1 - \Phi(2.6) = 0.0047. Conclusion: strong evidence the new alloy is stronger on average. Takeaway. A z of 2.6 is well beyond the cutoff, giving a p-value under 0.5%. The further the statistic pushes into the tail, the smaller the p-value and t…

Does the new alloy exceed 400 MPa tensile strength?

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