2. Inferences about the difference between two population means - Independent samples, population standard deviations known

Consider a pool of home mortgages. Prepayments of mortgages in the pool affect the mortgages' cash flow, so mortgage lenders, servicers, and investors all have an interest in predicting mortgage prepayments. Mortgages may be prepaid for a variety of purposes, including selling the home, taking cash out of the property to fund home improvements or other consumer expenditures, or refinancing the mortgage to change the monthly payment schedule.

Narrow your focus to mortgage prepayments that are made for the purpose of refinancing. If there were no costs to refinancing, you would refinance to reduce your monthly payments every time the current mortgage rate dropped below the rate on your mortgage. In actuality, however, there are costs to refinancing, such as points and closing fees. Therefore, the spread between the current mortgage rate and your own rate must be big enough to more than make up for the costs, or you wouldn't be interested in refinancing.

The economics of refinancing suggest that there should be no difference between the number of points on refinanced mortgages and mortgages that are not refinanced. Unlike the points paid on the new mortgage, points paid on the original mortgage are a sunk cost that should not be considered in the refinancing decision.

Define population 1 as mortgages that are refinanced, and define population 2 as mortgages that are not refinanced. Let equal the mean points paid on refinanced mortgages, and let equal the mean points paid on mortgages that are not refinanced. Similarly, let and equal the standard deviations of points paid for populations 1 and 2. Assume that = 0.94 and = 1.09.

In a study, professor Michael LaCour-Little selectedindependentrandom samples of mortgages that were refinanced and mortgages that were not refinanced, and he collected data on points paid. (Source: Michael LaCour-Little, "Another Look at the Role of Borrower Characteristics in Predicting Mortgage Prepayments,"Journal of Housing Research, Volume 10, Issue 1.)

For the sample drawn from refinanced mortgages, the sample size n = 356, and the sample meanxx = 0.69. For the sample drawn from mortgages that were not refinanced, the sample size n = 350, and the sample meanxx = 0.89. (Note:The sample means match those from the study, but the sample sizes have been reduced.)

The point estimate of - is .

In this study, the sampling distribution of x x - x x is approximated by a ____(ans options = t, standard normal, normal, uniform) distribution with ___ (ans options = a mean equal to the unknown value of h1-h2, a mean of 0.2, a mean of -0.2 ,a mean of 0) and a standard deviation ___ (ans options = of 1.09, 0.0767, that is unkown, of 0.94) .

Use the Distributions tool to help you answer the questions that follow.

The 95% confidence interval estimate of the difference between and is ____ to ___ .

You want to verify that there is no difference between the mean points paid on refinanced mortgages and mortgages that are not refinanced, as the economics of refinancing suggests. You test the hypothesis that there is no difference between the mean points paid.

The null and alternative hypotheses are formulated as:

a. H: - 0, Haa: - = 0

b. H: - = 0, Haa: - 0

c. H: x x - x x = 0, Haa: x x - x x 0

d. H: - 0, Haa: - > 0

The test statistic for the hypothesis test is____ .

The p-value is ____ .

A level of significance of = 0.05 is specified for the study. The null hypothesis is ____ . Therefore, you ____ conclude that there is a difference between the mean points on refinanced mortgages and mortgages that are not refinanced.