Question

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

Answer 1

SOLUTION:

From given data,

Data 13 A

H₀ : =80.0

H_a : 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: μ_x = 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: μ_x = 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

Please thumbs-up / vote up this answer if it was helpful. In case of any problem, please comment below. I will surely help. Down-votes are permanent and not notified to us, so we can't help in that case.