Question

A brochure inviting subscriptions for a new diet program states that
the participants are expected to lose over 22 pounds in five weeks. The statement
in the brochure will be accepted if the mean five-week weight loss of 100
randomly selected participants exceeds 24 pounds. Assume that the standard
deviation of the five-week weight losses is 12 pounds.
(a) Set up the null and alternative hypotheses.
(b) Compute the probability of Type I error for the indicated test.
(c) Compute the probability of Type II error for the indicated test when the true
mean weight loss, after five weeks under the diet, is 23 pounds.

Answer 1

here

(a) Null Hypothesis : H₀ : Mean five week weight loss which is expected to loose is not more than 22 pounds in five weeks. 22pounds

ALterntive Hypothesis : H_a : Mean five week weight loss which is expected to loose is more than 22 pounds in five weeks. > 22 pounds

(b) Here n = 100

standard deviation = 12 pounds

standard error of mean = 12/sqrt(100) = 1.2 pounds

critical range is for more than 24 pounds

Pr(TYpe I error) = Pr( > 24 pounds; 22 pounds ; 1.2 pounds)

Z = (24 - 22)/1.2 = 1.6667

Pr(Type I error) = Pr(Z >1.6667) = 0.0478

Type I error of 0.0478

(c) Here if real weight loss is 23 pounds

so here

Pr(TYpe II error) = Pr( < 24 pounds ; 23 pounds ; 1.2 pounds)

Z = (24 - 23)/1.2 = 0.8333

Pr(Type II error) = Pr(Z < 0.8333) = 0.7977