4 . Comparing two population means (known sigmas) 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 dosing 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 compared to mortgages that aren't refinanced, refinanced mortgages are larger. The larger the mortgage, the greater the gain from refinancing. Define population 1 as mortgages that are refinanced, and define population 2 as mortgages that are not refinanced. Let pi equal the mean loan size of refinanced mortgages, and let uz equal the mean loan size of mortgages that are not refinanced. Similarly, let on and oz equal the standard deviations of loan sizes for populations I and 2. Assume that ci = 1.21 and oz = 0.95. In a study, professor Michael LaCour-Little selected independent random samples of mortgages that were refinanced and mortgages that were not refinanced, and he collected data on loan sizes. (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, = 115, and the sample mean $1 = 1.62. For the sample drawn from mortgages that were not refinanced, the sample size n, = 100, and the sample mean $2 = 1.23. Histograms of the sample data suggest that the populations are not extremely nonnormal. (Note: The sample means match those from the study, but the sample sizes have been reduced.) The point estimate of ul - uz Is In this study, the sampling distribution of X1 - $2 Is approximated by a distribution with and a standard deviation Use the Distributions tool to help you answer the questions that follow. Standard Normal Distribution Mann = 00 Standard Deviation = 1.0 0 0 0 AAN 2 The 90% confidence Interval estimate of the difference between wi and #2 Is LCL = to UCL = You want to determine whether refinanced mortgages have a larger mean loan size than mortgages that are not refinanced, as the economics of refinancing suggests. You test the hypothesis that there is no difference between the mean loan sizes. The null and alternative hypotheses are formulated as: D H : HI - 12 2 0, H,: HI - 12 = 0 OH : HI - 12 = 0, H,: HI - 12 > 0 OH: (1 - 12 2 0, H.: HI - p2 0 The test statistic for the hypothesis test Is The p-value Is A level of significance of a = 0.10 Is specified for the study. The null hypothesis Is . Therefore, you conclude that there is enough evidence to Infer that refinanced mortgages have a larger mean loan size than mortgages that are not refinanced