EXAMPLE 11 Pooled variance test

image Bank Loans

Another indicator of financial health that banks look at when evaluating loan applicants is the debt-to-income ratio, which is defined as follows.

For example, if you owed $5000 on your credit card and $10,000 on your car, and your annual income was $50,000, then your debt-to-income ratio would be .

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Here, we examine whether the mean debt-to-income ratio differed between those approved and those denied the bank loans. Samples of size 100 were obtained from each group, with the summary statistics shown in Table 11.

Table 10.23: Table 11 Summary statistics for debt-to-income ratios of loan applicants who were approved and those who were denied the loan
Loan status Sample size Sample mean
debt-to-income
ratio		 Sample standard
deviation		 Population mean
debt-to-income
ratio
Approved
(sample 1)
Denied
(sample 2)

Use the critical-value method for the pooled variance test to test whether the population mean debt-to-income ratio for those approved is less than that of those who were not approved for the loan. Assume and use level of significance .

Solution

  • Step 1 State the hypotheses.

    where and represent the population mean debt-to-income ratio for those approved and those denied the loan, respectively.

  • Step 2 Find .

    The degrees of freedom for the pooled variance test equals . Because is not in the table, we use the next lower value instead, obtaining the critical value . Reject if .

  • Step 3 Calculate and .

    Plugging this value into the following formula for the test statistic, we obtain

  • Step 4 Conclusion and interpretation.

    The test statistic is less than the critical value . Therefore, we reject . There is evidence that the population mean debt-to-income ratio for those whose loan was approved is less than that for those whose loan was not approved.

NOW YOU CAN DO

Exercises 21 and 22.