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Loans 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. We can 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. This system is reasonably reliable, but, of course, sometimes there are mistakes. a) When a person defaults on a loan, which type of error did the bank make? b) Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? c) Suppose the bank decides to lower the cutoff score from 250 points to \(200 .\) Is that analogous to choosing a higher or lower value of \(\alpha\) for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?

Step by step solution

01

Identify Type I Error

In the context of hypothesis testing, rejecting a true null hypothesis constitutes a Type I error. Here, the null hypothesis is that the applicant will repay the loan. If a person defaults on a loan, the null hypothesis was accepted, even though it should have been rejected. Therefore, no Type I error occurs when a person defaults; this is actually a success of rejecting a false null hypothesis.

02

Identify Type II Error

A Type II error occurs when the null hypothesis is false, but it is not rejected. In this case, when the bank denies a loan to someone who would have repaid it, they fail to reject a false null hypothesis. Hence, this is an example of a Type II error.

03

Determine the Effect of Lowering the Cutoff Score

Lowering the cutoff score from 250 to 200 points makes it easier for applicants to qualify for a loan. This is analogous to choosing a higher value of \( \alpha \) in hypothesis testing, which makes it more likely to reject the null hypothesis.

04

Analyze Error Impact of Cutoff Change

Reducing the cutoff score can increase the probability of a Type I error (approving loans for those who default) because more applicants will qualify. Conversely, it decreases the probability of a Type II error (rejecting good applicants), because more people are approved.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

In the world of hypothesis testing, a Type I error is considered one of the classic pitfalls. Put simply, it occurs when the null hypothesis is incorrectly rejected. In the context of our banking scenario, the null hypothesis is that the applicant will repay the loan. For a Type I error to occur here, the bank would need to deny a loan, believing the applicant is a bad risk, only to later find that the applicant would have been perfectly capable of repaying. This mistaken rejection means missing out on potential profit from interest, which the bank believed would be risky.

• This type of error leads to unnecessarily denying applicants.
• It represents a cautious approach but can mean lost opportunities.

Type II Error

Type II errors often sneak up in decision-making processes, especially in lending scenarios. They happen when the bank fails to reject a false null hypothesis. In other terms, it means the bank accepts the null hypothesis that an applicant will repay their loan, when in reality, they will not. For example, if the denial of a loan was meant for someone who actually would have repaid it, then the bank missed out on a good loan, and this counts as a Type II error.

• This error entails being overly lenient or optistically confident.
• Such errors can lead to defaults, potentially harming the bank's finances.

Type II errors stress the need for careful judgment to avoid misplaced confidence in an applicant's ability to repay.

Loan Approval Criteria

Creating the right loan approval criteria is like striking a delicate balance. Banks generally use various metrics in a point system to screen potential borrowers. This includes evaluating income levels, credit histories, and current debts to predict repayment capacity.

• Higher scores indicate a higher likelihood of repayment.
• The bank sets a cutoff value to filter out high-risk applications.

Such criteria ensure a structured approach to decision-making, allowing banks to maximize their return on loans by minimizing risk. A thoughtful balance in criteria can protect against both Type I and Type II errors, ensuring proper loan allocation.

Cutoff Score in Banking

A cutoff score acts as a critical point in a bank's lending process. It separates applicants deemed suitable for loans from those considered too risky.Adjusting the cutoff score has a direct impact on the type of errors the bank is likely to make. Lowering the score means:

• More applicants will qualify for loans, raising the possibility of overlooking a person who defaults (higher Type I error).
• It reduces the chance of missing out on reliable payers, hence decreasing the chance of a Type II error.

Analogous to adjusting \(\alpha\) in hypothesis testing, it involves trade-offs between accepting more applicants and increasing potential risk.Finding the optimal cutoff minimizes errors while balancing the bank's risk appetite with potential earnings.