With regard to Data 13A, (a) if a=.05 and utrue=84.0, what is B? What is the power of the test? (b) If a=.05 and utrue=84.0, but Ha:ux>80.0, what is B? What is the power of the test? Data 13A H0:ux=80.0 Ha:ux does not equal 80.0 rx=20.0 n=100
SOLUTION:
From given data,
Data 13 A
H0 : =80.0
Ha : 80.0
= 20.0
n =100
(a) If α = .05 and μtrue = 84.0, what is β? What is the power of the test?
Since test is two tailed so critical values of z is -1.96 and 1.96. Therefore critical values of are
Left critical value:
z = (critical - ) / ( /sqrt{n} )
-1.96 = (critical - 80 ) / ( 20/sqrt{100} )
critical =76.08
Right critical value:
z = (critical - ) / ( /sqrt{n} )
1.96 = (critical - 80 ) / ( 20/sqrt{100} )
critical =83.92
That is we will reject the H0 when mean is left to 76.08 or mean is right to 83.92. Power of the test is the probability of rejecting Ho: μx = 80.0 when μtrue = 84.0. So the probability that is less than
critical =76.08 , when μtrue = 84.0 will be
P( z < {76.08-84} / {20/sqrt{100}})=P(z < -3.96) = 0
And the probability that is greater than critical =83.92 , when μtrue = 84.0 will be
P ( z > {83.92-84}/{20/sqrt{100}})=P(z > -0.04) = 0.516
So power of the test is
power=0+0.516=0.516
And
=
1-power=1-0.516=0.484
(b) α = .05 and
μtrue = 84.0, but Ha: μx > 80.0, what is β?
What is the power of the test?
Since test is right tailed so critical values of z is 1.645. Therefore critical values of are
Left critical value:
z = (critical - ) / ( /sqrt{n} )
1.645 = (critical - 80 ) / ( 20/sqrt{100} )
critical = 83.29
That is we will reject the H0 when mean is right to 83.29. Power of the test is the probability of rejecting Ho: μx = 80.0 when μtrue = 84.0. So the probability that is greater than
critical =83.29 , when μtrue = 84.0 will be
P( z > {83.29-84} / {20/sqrt{100}})=P(z > -0.355) = 0.6387
So power of the test is
power = 0.6387
And
= 1-power=1-0.6387=0.3613
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