A certain size of bag is designed to hold 25 pounds of potatoes. A farmer fills such

Chapter 8, Problem 8.5-1

(choose chapter or problem)

A certain size of bag is designed to hold 25 pounds of potatoes. A farmer fills such bags in the field. Assume that the weight \(X\) of potatoes in a bag is \(N(\mu, 9)\). We shall test the null hypothesis \(H_{0}: \mu=25\) against the alternative hypothesis \(H_{1}: \mu<25\). Let \(X_{1}, X_{2}, X_{3}, X_{4}\) be a random sample of size 4 from this distribution, and let the critical region C for this test be defined by \(\bar{x} \leq 22.5\), where \(\bar{x}\) is the observed value of \(\bar{X}\).

(a) What is the power function \(K(\mu)\) of this test? In particular, what is the significance level \(\alpha=K(25)\) for your test?

(b) If the random sample of four bags of potatoes yielded the values \(x_{1}=21.24, x_{2}=24.81, x_{3}=23.62\), would your test lead you to accept or reject \(H_{0}\)?

(c) What is the p-value associated with \(\overline{\boldsymbol{x}}\) in part (b)?