A certain size of bag is designed to hold 25 pounds of

Chapter 8, Problem 1E

(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 \(x_{1}=21.24, x_{2}=24.81, x_{3}=23.62\), and \(x_{4}=26.82\), would your test lead you to accept or reject \(H_{0}\) ?

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

Equation Transcription:

 

 

 

 

 

 

Text Transcription:

X

N(mu,9)

H_0:=25

H_1:<25

X_1,X_2,X_3,X_4

Bar x < or = 22.5

X

Bar X

K(mu)

Alpha =K(25)

x_1=21.24, x_2=24.81, x_3=23.62

x_4=26.82

H_0

p