A union of restaurant and foodservice workers would like to estimate this year's mean hourly wage for foodservice workers in the U. S. Last year's mean hourly wage was $8.08, and there is reason to believe that this year's value is greater than last year's. The union decides to do a statistical test to see if it can be concluded that the mean has increased. The union chooses a random sample of 100 wages from this year. Suppose that the population of hourly wages of foodservice workers in the U. S. has a standard deviation of $1.18 and that the union performs its hypothesis test using the 0.1 level of significance. Based on this information, answer the questions below. Carry your intermediate computations to at least four decimal places, and round your responses as indicated.(If necessary, consult a list of formulas.)
a. What are the null and alternative hypotheses that the union should use for the test?
b. What is the probability that the union rejects the null hypothesis when, in fact, it is true? Round your response to at least two decimal places.
c. Assuming that the actual value is $8.28, what is the power of the test? Round your response to at least two decimal places.
d. Suppose that the union decides to perform another statistical test using the same population, the same null and alternative hypotheses, and the same level of significance, but for this second test the union chooses a random sample of size 75 instead of a random sample of size 100. Assuming that the actual value is $8.28, how does the probability that the union commits a Type II error in this second test compare to the probability that the union commits a Type II error in the original test?