Question

In a news story distributed by the Washington Post, Lew Sichelman reports that a substantial fraction of mortgage loans that go into default within the first year of the mortgage were approved on the basis of falsified applications. For instance, loan applicants often exaggerate their income or fail to declare debts. Suppose that a random sample of 1,000 mortgage loans that were defaulted within the first year reveals that 410 of these loans were approved on the basis of falsified applications.

(a) Find a point estimate of and a 95 percent confidence interval for p, the proportion of all first-year defaults that are approved on the basis of falsified applications. (Round your answers to 4 decimal places.) The 95 percent confidence interval is
(b) Based on your interval, what is a reasonable estimate of the minimum percentage of all first-year defaults that are approved on the basis of falsified applications?

Answer 1

To find the point estimate and a 95% confidence interval for the proportion of defaults approved on the basis of falsified applications, divide the number of defaults approved on falsified applications by the total number of defaults. The point estimate is 0.41 with a 95% confidence interval of (0.3937, 0.4263).

Step-by-step explanation:

To find the point estimate and a 95% confidence interval for the proportion of first-year defaults that were approved on the basis of falsified applications, we use the formula:

Point estimate = (Number of defaults approved on falsified applications) / (Total number of defaults)

Confidence interval = Point estimate ± (Z-score)(Standard error)

Using the given data, the point estimate is 410/1000 = 0.41. The Z-score for a 95% confidence level is approximately 1.96. The standard error can be calculated using the formula: Standard error = √((Point estimate)(1 - Point estimate) / Sample size) = √((0.41)(1 - 0.41) / 1000) ≈ 0.0155.

Plugging in the values, the confidence interval is approximately (0.3937, 0.4263). Therefore, the point estimate of the proportion of first-year defaults approved on the basis of falsified applications is 0.41, and with 95% confidence, we can estimate that the true proportion falls between 0.3937 and 0.4263.