Determine the power for the following test of hypothesis H0

Determine the power for the following test of hypothesis.
H0 : = 950 vs. H1 : 950, given that = 1,000, = 0.10, = 200, and n = 25.
A. 0.6535
B. 0.5062
C. 0.4938
D. 0.3465

Solution

First, we get the z score from the given left tailed area. As

The left tailed area of the right endpoint of the test is 0.95.          
          
Left tailed area =    0.95      
          
Then, using table or technology,          
          
z =    1.644853627      
          
As x = u + z * s / sqrt(n)          
          
where          
          
u = mean =    950      
z = the critical z score =    1.644853627      
s = standard deviation =    200      
n = sample size =    25      
Then          
          
x = critical value =    1015.794145

This is the right critical value for the hypothesis test. Now, for the probability of getting a value farther (right tailed area):      

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    1015.794145      
u = mean =    1000      
n = sample size =    25      
s = standard deviation =    200      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    0.394853625      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   0.394853625   ) =    0.346475458 = 0.3465 [ANSWER, B]