In a city, a radio program hosted by John has long been very popular. John has just been retired and Jack will be the new host of the program. Let be the proportion of residents in the city who are regular listeners of the program. It is known that =0.5 before John retired. The radio company conducts a statistical test on 1000 interviewees to examine whether the proportion decreases after John's retirement. If so, the company will fire Jack and cancel the program. Otherwise, no change will be made.

a. Write down the null hypothesis and alternative hypothesis in this case.

b. What are the type I error and type II error in this case?

c. Before conducting the statistical test, the company is not willing to fire Jack and cancel the program. Given there are three choices for :0.1,0.05,0.01, which one should the company use for the test? Explain your answer briefly.

d. Given the sample size is unchanged, if the company reduce the significance level, what will the probability of committing the type II error change?

e. Suppose the company finds that the sample proportion is not significantly smaller than 0.5. Then the company concludes that there is enough evidence not to fire Jack and continue the program. Is the conclusion correct? Explain your answer briefly.