An inspector of flow metering devices used to administer flu

An inspector of flow metering devices used to administer fluid intravenously will perform a hypothesis test to determine whether the mean flow rate is different from the flow rate setting of 200 milliliters per hour. Based on prior information the standard deviation of the flow rate is assumed to be known and equal to 14 milliliters per hour. For the sample size n = 50, and = 0.05, find the probability of a type II error if the true mean is 205 milliliters per hour. Round the final answer to four decimal places (e.g. 98.7654).
Give your answer.
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Solution

Set Up Hypothesis
Null, H0: U=200
Alternate, H1: U!=200
Test Statistic
Population Mean(U)=200
X(Mean)=205
Standard Deviation(S.D)=14
Number (n)=50
Critical Value
The Value of |Z | at LOS 0.05% is 1.96
Since our test is two tailed
Reject Ho if Zo < -1.96 , or Zo>1.96
Reject Ho if x-200/(14/Sqrt(50) < - 1.96, or x-200/(14/Sqrt(50) > 1.96
Reject Ho if x-200/(1.9798) < - 1.96, or x-200/(1.9798) > 1.96
Reject Ho if x < 196.119 , or x > 203.880

Implies do n\'t reject Ho if 196.119 < = X < = 203.880

Suppose the true mean is 205
P( Type II Error) = P( Don\'t Reject Ho| H1 is True )
               = P[ (196.119 - 205) / 14/Sqrt(50) < x-U/(s.d/Sqrt(n)) < (203.880 - 205) / 14/Sqrt(50)) ]
               = P[ -4.4855 < x-U/(s.d/Sqrt(n)) < -0.56568 ]
               = P[ (Z<=-0.5657) - (Z<=-4.4855) ]
               = 0.2858 - 0
               = 0.2858