Question

Before lending someone​ money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the​ applicant, totaling points they award for the​ person's income​ level, credit​ history, current debt​ burden, and so on. The higher the point​ total, the more convinced the bank is that​ it's safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. Think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid​ loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the​ person's score falls below the minimum cutoff will the bank reject the null and deny the loan. Complete parts a through c below.

a) In this context, what is meant by the power of the test?

A. The power is the probability that the bank approves a loan that will be repaid.

B. The power is the probability that the bank denies a loan that would have been repaid.

C. The power is the probability that the bank approves a loan that will not be repaid.

D. The power is the probability that the bank denies a loan that would not have been repaid.

b) What could the bank do to increase the power?

A. The bank could hire additional loan officers to assess each​ applicant's information.

B. The bank could lower the cutoff score.

C. The bank could raise the cutoff score.

D. The bank could scrap the point system.

c) What is the disadvantage of taking the action in part b)?

A. A larger number of untrustworthy people would have their loans​ approved, and the bank would lose money from those unpaid loans.

B. The bank would have to spend more money on the additional loan officers.

C. The bank would have to spend additional time and money developing a new system.

D. A larger number of trustworthy people would be denied​ credit, and the bank would miss the opportunity to collect interest on those loans.

Answer 1

Assume the given scenerio. Since the bank makes its profit from the interest collected on repaid​ loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the​ person's score falls below the minimum cutoff will the bank reject the null and deny the loan.

Let be the total point of a person. Then if be the cutoff value,then

The null hypotheis is .

Under H0 , Bank give money to person as he will repay the loan and under H1, bank will not give money to person as he wolud not return the loan

(a) The power of a test is defined as the probability of rejecting the null when it is false or alternative is true. Hence for the given test, power of test will be the probability that bank denies the loan that would not have been repaid.

So option (D) is true.

(b) To increase the power of the test, Bank should increase the cutoff value of the totalling point. As hiring additional loan officers to assess each​ applicant's information does not going to help bank. Also lowering the cutoff value makes it attempting to increase the Type II error consequently decreses the power. Scraping the point system would not going to increase the power as the bank have to make another test plan.

Hence option (C) will be right option.

(c) Option (A)- A larger number of untrustworthy people would have their loans​ approved, and the bank would lose money from those unpaid loans is incorrect as it is consequence of lowering the cutoff value.

Option (B)- The bank would have to spend more money on the additional loan officers , is also incorrect as it is resultant of hiring additional loan officers.

Option (C)- The bank would have to spend additional time and money developing a new system is incorrect as it is resultant of scraping the point system.

So option (D) is correct option. Because now increaseing the cut-off value for totalling point to increase the power would result in rejecting more person to give them loan. This results in rejecting some trustworthy person and the bank would miss the opportunity to collect interest on those loans.