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2. Comparing two population means (independent samples, sigmas known) Consider a podl 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 consuner expenditures, or refinancing the mortgage to change the monthly payment schedule. Narrow your focus tomortgage 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 refinanding, such as points and dlosing fees. Therefore, the spread between the current mortgage rate and your own rate must be big enaugh to more than make up for the costs, or you wouldnt be interested in refinanding. The economics of reinancing suggest that compared to mortgages that arent refinanced, refinanced mortgages have borrowers with higher FICO scores. Borowers with good credit histories (high credit scores) pay lower costs when they refinance. Define population 1 as mortgages that are refinanced, and define population 2 as mortgages that are not refinanced μι equal te mean FICO score of borrowers who refinance their mortgages, and let equal te mean FICO score of borrowers who dont refinance. Similarly, let and 02 equal te standard deviations of borrower FICO scores for populations 1 and 2. Assume that σ1 = 45 and 02 = 66. 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 borrower FICO scores. (Source: Michael LaCour-Litte, "Another Look at the Role of Borrower Characteristics in Predicting Mortgage Prepayments,"Joumal of Housing Ressarch, Volume 10, Issue 1.) For the sample drown from refinanced mortgages, the sample size n-124, and the sample mean &-740, For the sample drawn from mortgages that were nat refinanced, the sample size nz 124, and the sample mean 720. (Note: The sample means match those from the study, but the sample sizes have been reduced) The point estimate of μ-Hz is In this study, the sampling distribution of R1 8z is approximated by a distribution with and a standard deviation Use the Distributions tool to help you answer the questions that follow Standard Nonmaí Distribution Mesn 0.0 Standard Deviation 1.0 90% confidence interval for the difference between μ1 and μ2 is to You want to determine whether the mean FICO score of the borrower is higher for refinanced mortgages than for mortgages that are not refinanced, as the economics of refinancing suggests. You test the hypothesis that there is no difference between the mean FICO scores The null and alternate hypotheses are formulated as The best statistic for the hypothesis best is The P-value is A level of significance of α . .10 is speched for the study. The null hypothesis is Therefore, You condlude that the mean FICO score of the borrower is higher for refinanced mortgages than for mortgages that are not refinanced A level of significance of a.10 is specified for the study. The null hypothesis is Therefore, you conclude that the mean FICO score of the borrower is higher for refinrejectedthan for not rejected mortgages that are not refinanced.

Answer 1

Point estimate of $\mu$1-$\mu$2 is 740-720=20

The sampling distribution of x1bar-x2bar is approximated by a standard normal distribution with mean equal to unknown value of $\mu$1-$\mu$2 and a standard deviation of 45^2+65^2=6250

The critical value for α=0.1 is z c = z_{1-alpha/2} = 1.645

The critical value for α 0.1 is - 1.645. The corresponding confidence 2 1-α/ interval is computed as shown below: n1 2 n1 m2 452 652 124 124 452 652 124 124 740 720- 1.645 ,740 - 720 + 1.645 (8.322, 31.678)

The null and alternate hypotheses are

The last option is correct.

The test statistic is:

(3) Test Statistics The z-statistic is computed as follows: 740-720 452/124+652/124-2.817 Z- /nutơ /M-

p value:

The p-value is p = 0.0024

p=0.0024<0.10, it is concluded that

the null hypothesis is rejected.

Therefore you can conclude that the mean FICD score of the borrower is higher for refinanced mortgages than for mortgages that are not refinanced.