For the following right-tailed test, given H0 : µ ≤ 135 vs H1 : µ > 135 with n = 59, α = 0.05, s = 10, and = 140. Find the observed power

Question

a. 0.99

b. 0.95

c. 0.01

d. 0.05

Answer 1

In Hypothesis Testing:

a Type I Error is committed when one fails to accept a true Null Hypothesis; 

a Type II Error is committed when one fails to reject a false Null Hypothesis.

Here: 

population mean is μ = 135;

sample size is n = 59;

significance level is α = 0.05;

sample standard deviation is s = 10;

and sample mean is X_bar = 140.

Construct (X_bar − μ)/[s/n^0.5] as (140 − 135)/[10/59^0.5]

which reduces to 0.5[59^0.5] or 3.840572874.

For significance level of α = 0.05, with a right-tailed test, 

one takes a critical value of z (or z_c) equal to 1.645.

3.840572874 falls well past 1.645 into the right-hand region of rejection.

For

H₀: μ ≤ 135 &

H₁: μ > 135,

one then fails to accept H₀ and concludes that μ exceeds 135.

The area to the right of 3.840572874 under the standard normal curve is given by

β equal to  1 − (0.5 + 0.4999386262)  or 6.13737711 × 10^-5.

The power of the test here is given by 1 minus the probability of failure to reject a false Null Hypothesis or 1 − Pr(Type II Error).

The power of a test is a measure of the test's efficiency in recognizing true hypotheses as valid and false hypotheses as flawed and here amounts to 1 − β or 1 − 6.13737711 × 10^-5 or 0.9999386262 equivalent to 0.99, which corresponds to Choice A.