The calibration of a scale is checked by weighing a standard \(10 \mathrm{~g}\) weight 100 times. Let \(\mu\) be the population mean reading on the scale, so that the scale is in calibration if \(\mu=10\). A test is made of the hypotheses \(H_{0}: \mu=10\) versus \(H_{1}: \mu eq 10\).

Consider three possible conclusions: (i) The scale is in calibration. (ii) The scale is out of calibration.

(iii) The scale might be in calibration.

a. Which of the three conclusions is best if \(H_{0}\) is rejected?

b. Which of the three conclusions is best if \(H_{0}\) is not rejected?

c. Is it possible to perform a hypothesis test in a way that makes it possible to demonstrate conclusively that the scale is in calibration? Explain.