Are the steel rods being cut too long?

Question

A cutting machine is set to produce steel rods of length 120 mm. The machine's long-run standard deviation is known from calibration to be $\sigma = 1.5$ mm. An inspector measures 25 rods and gets a mean of 120.75 mm.

Test at the 5% level (two-sided) whether the machine is off target, and report the z-statistic.

Accepted Answer

The population standard deviation is known, so use a z-test. Hypotheses: H0: \mu = 120 versus H1: \mu 
eq 120. Standard error: 	ext{SE} = \frac{1.5}{\sqrt{25}} = \frac{1.5}{5} = 0.30 	ext{ mm}. Test statistic: z = \frac{120.75 - 120}{0.30} = \frac{0.75}{0.30} = 2.5. Since 2.5 1.96, reject H0. The two-sided p-value is 2\,[1 - \Phi(2.5)] = 2(0.0062) \approx 0.012. Conclusion: the rods are running significantly long, by about 0.75 mm on average. Takeaway. A small absolute drift (0.75 mm) can still be highly significant when the process is tight (small \sigma) and the sample is decent, because…

Are the steel rods being cut too long?

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