DATA 13A Ho x 800 x 200 Ha x 800 n 100 In regard to DATA 1

DATA 13A

Ho: x = 80.0 x = 20.0   

Ha: x 80.0 n =100

In regard to DATA 13A...

(a) If = .05 and true = 83.0, what is ? What is the power of the test?

(b) If = .01 and true = 83.0, what is ? What is the power of the test?

Solution

a) std. error = 20/ sqrt(100) = 2... z -value for 0.95 for a 2 sided test is + or - 1.96...
z = (x-bar - x) / std.error or, 1.96 = ( x-bar - 80 ) / 2 i.e, x-bar(upper limit) = 83.92...
or, - 1.96 = ( x-bar - 80) /2 i.e, x-bar(lower limit) = 76.08......

so, beta = p( x-bar> 83.92 when true = 83.0 ) + p( x-bar< 76.08 when true = 83.0)
i.e, beta = p ( z > ( 83.92-83) / 2 ) + p ( z < (76.08 - 83 ) / 2)
i.e, beta = 0.3227581+ 0.0002700877

so, power = 1- (0.3227581+ 0.0002700877) =  0.6769718

b) z= + or - -2.575829...so, x-bar(upper limit) = 85.15...
x-bar(lower limit) = 74.85...
beta= p( z > 1.075 ) + p ( z < -4.08) = 0.1563598
power = 1- beta = 0.8436402....