For the following right-tailed test, given H0 : µ ≤ 135 vs H1 : µ > 135 with n = 39, α = 0.05, s = 10, and = 140. Find the observed power

Question

a. 0.81

b. 0.19

c. 0.93

d. 0.07

Answer 1

At alpha=0.05 for a right tailed test, Ho will be rejected when the t-statistic is equal or greater than a critical t value of 1.686.

you can calculate the x-bar at which the Ho will be reject using the formula for the t-statistic.

t = (x-bar - uo)/(s/sqrt(n)) or rearranged x-bar = uo + t * s/sqrt(n)

substituting in values x-bar = 135 + 1.686 *10/sqrt(39) = 137.7

To determine the power we now need to see where 137.7 fits in the actual distribution with a mean of 140 and s=10 with n=39

the t-statistic an x-bar of 137.7 with a mean of 140, s=10 and n=39 is (137.7-140)/(10/sqrt(39)) = -1.436

Beta, the probability of making a type II error is the area to the left of that t-statistic with a df of 38.

Using my TI-83 calculator tcdf(-99999,-1.436,38) = 0.080 = beta. The power of the test is 1 - beta.

or 1-0.08 = 0.92 So the best answer given here is 0.93.