Refer to Problem 26. Suppose that you decide to use a z-test with a significance level of 0.10 and a sample size of 25. Assume that σ = 9 gallons.

a. Determine the probability of a Type I error.

b. If last year’s mean beer consumption was 25 gallons, identify the distribution of the variable x¯, that is, the sampling distribution of the mean for samples of size 25.

c. Use part

(b) to determine the probability, β, of a Type II error if in fact last year’s mean beer consumption was 25 gallons.

d. Repeat parts

(b) and

(c) if in fact last year’s mean beer consumption was 20.5 gallons, 21 gallons, 21.5 gallons, 22 gallons, 22.5 gallons, 23 gallons, and 23.5 gallons.

e. Use your answers from parts

(c) and

(d) to construct a table of selected Type II error probabilities and powers.

f. Use your answer from part

(e) to construct the power curve.

Using a sample size of 50 instead of 25, repeat g. part (b). h. part (c). i. part (d).

j. part (e). k. part (f).

l. Compare your power curves for the two sample sizes and explain the principle being illustrated.

Problems 34 and 35 each include a normal probability plot and either a frequency histogram or a stem-and-leaf diagram for a set of sample data. The intent is to use the sample data to perform a hypothesis test for the mean of the population from which the data were obtained. In each case, consult the graphs provided to decide whether to use the z-test, the t -test, or neither. Explain your answer.