Is the coffee machine off its 250 ml target?

Question

A vending machine is set to pour 250 ml of coffee per cup. From years of maintenance logs, the pour-to-pour standard deviation is known to be $\sigma = 8$ ml. A technician samples 36 cups and finds a mean of 246 ml.

Run a one-sample z-test at the 5% level (two-sided): is the machine off target, and what is the z-statistic?

Accepted Answer

Because the population standard deviation is known, the correct tool is a z-test. Hypotheses: H0: \mu = 250 (on target) versus H1: \mu 
eq 250 (off target). The standard error of the sample mean is 	ext{SE} = \frac{\sigma}{\sqrt{n}} = \frac{8}{\sqrt{36}} = \frac{8}{6} = 1.333 	ext{ ml}. The test statistic is z = \frac{\bar x - \mu0}{	ext{SE}} = \frac{246 - 250}{1.333} = \frac{-4}{1.333} = -3.0. The two-sided 5% critical values are \pm 1.96. Since -3.0 = 3.0 1.96, we reject H0. The two-sided p-value is 2\,[1 - \Phi(3.0)] = 2(0.00135) \approx 0.0027, far below 0.05. Conclusion: there is str…

Is the coffee machine off its 250 ml target?

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