Default Rates on Bank Loans. Carl Allen and Norm Nixon are...

Default Rates on Bank Loans. Carl Allen and Norm Nixon are two loan offi- cers at Brea Federal Savings and Loan Bank. The bank manager is interested in comparing the default rate on the loans approved by Carl to the default rate on the loans approved by Norm. In the sample of loans collected, there are 60 loans approved by Carl (9 of which defaulted) and 80 loans approved by Norm ( 7 of which defaulted). a. State the hypothesis test that the default rates are the same for the two loan officers. b. What is the sample default proportion for Carl? For Norm? c. Use a .05 level of significance. What is the $p$ -value and what is your conclusion?

Default Rates on Bank Loans. Carl Allen and Norm Nixon are two loan offi-
cers at Brea Federal Savings and Loan Bank. The bank manager is interested in comparing the default rate on the loans approved by Carl to the default rate on the loans approved by Norm. In the sample of loans collected, there are 60 loans approved by Carl (9 of which defaulted) and 80 loans approved by Norm ( 7 of which defaulted).
a. State the hypothesis test that the default rates are the same for the two loan officers.
b. What is the sample default proportion for Carl? For Norm?
c. Use a .05 level of significance. What is the $p$ -value and what is your conclusion?

Key Concepts

Hypothesis Testing for Two Proportions

This concept involves comparing the proportions of a particular outcome in two independent groups. It requires the formulation of a null hypothesis that assumes no difference between the groups and an alternative hypothesis that indicates a difference. This testing framework is used to evaluate whether any observed differences in sample proportions can be attributed to random chance or indicate a true difference in the underlying populations.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) asserts that there is no effect or no difference—in this case, that the default rates for the two loan officers are equal. The alternative hypothesis (Ha) posits that there is a difference in the default rates. Clearly defining these hypotheses is essential for setting up the test and making statistical inferences.

Sample Proportion

A sample proportion is calculated as the number of observed events (such as defaults) divided by the total sample size. It serves as an estimate of the population proportion and is critical for comparing outcomes between groups. The sample proportion provides the baseline figures necessary for computing test statistics in comparative studies.

Significance Level

The significance level, often set at 0.05 in many studies, defines the threshold probability for rejecting the null hypothesis. It represents the maximum allowable probability of committing a Type I error—rejecting a true null hypothesis. The significance level is used to determine whether the p-value of the test statistic is sufficiently low to conclude that the observed difference is statistically significant.

P-value

The p-value quantifies the probability of observing the data, or something more extreme, if the null hypothesis were true. A p-value lower than the predetermined significance level provides evidence against the null hypothesis, leading to its rejection. It is a crucial component in decision-making during hypothesis testing, as it directly informs whether the observed differences are statistically significant.

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