Let X1,X2, . . . ,Xn be a random sample from the normal

Chapter 8, Problem 6E

(choose chapter or problem)

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from the normal distribution \(N(\mu, 9)\). To test the hypothesis \(H_{0}:\) \(\mu=80\) against \(H_{1}: \mu \neq 80\), consider the following three critical regions: \(C_{1}=\left\{\bar{x}: \bar{x} \geq c_{1}\right\}, C_{2}=\left\{\bar{x}: \bar{x} \leq c_{2}\right\}\), and \(C_{3}=\left\{\bar{x}:|\bar{x}-80| \geq c_{3}\right\}\).

(a) If \(n=16\), find the values of \(c_{1}, c_{2}, c_{3}\) such that the size of each critical region is \(0.05\). That is, find \(c_{1}, c_{2}\), \(c_{3}\) such that

\(\begin{aligned}0.05 &=P\left(\bar{X} \in C_{1} ; \mu=80\right)=P\left(\bar{X} \in C_{2} ; \mu=80\right) \\&=P\left(\bar{X} \in C_{3} ; \mu=80\right)\end{aligned}\)

(b) On the same graph paper, sketch the power functions for these three critical regions.

Equation Transcription:

 

  

 

Text Transcription:

X_1,X_2,…,X_n  

N(mu,9).  

H_0: mu=80  

H_1:munot = 80

C_1={bar x: bar x > or = c_1,C_2={bar x: bar x < or = c_2}

C_3={ bar x:| bar x -80| > or = c_3}.

n=16

c_1,c_2,c_3  

0.05.  

0.05  =P( bar X in C_1;mu=80)=P(bar X in C_2; mu=80)  =P( bar X in C_3;=80)