Calculate the probability of a Type II error for the followi

Calculate the probability of a Type II error for the
following test of hypothesis, given that = 203.
H0: = 200
H1: 200
= .05, = 10, n = 100

Solution

Set Up Hypothesis
Null, H0: U=200
Alternate, H1: U!=200
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
Since test is two tailed,
Reject Ho if Zo<-1.96 OR Zo>1.96
Reject Ho if x-200/(10/Sqrt(100)<-1.96 OR x-200/(10/Sqrt(100)>1.96
Reject Ho if x<-1.96+200 OR x>1.96+200
Reject Ho if x<198.04 OR x>201.96
Implies, don\'t reject if 0 if 198.04 < X < 201.96
Suppose the true mean is 203.
Probability of type II error,
P(type II Error )=P(Don’t reject H0 | H1 is true)
               = P(198.04 < X < 201.96 | u = 203)
               = P ( 198.04 - 203/(10/Sqrt(100) < x-U/(s.d/Sqrt(n)) < 201.96-203/(10/Sqrt(100) )
               = P ( - 4.96 < x-U/(s.d/Sqrt(n)) < -1.04 )
               = P(- 4.96 < Z < -1.04 )
               = 0.1492