2. Comparing two population means (independent samples, sigmas known) Aaa Aa Con

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2. Comparing two population means (independent samples, sigmas known) Aaa Aa 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 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 H1 equal the mean loan size of refinanced mortgages, and let H2 equal the mean loan size of mortgages that are not refinanced. Similarly, let 01 and o2 equal the standard deviations of loan sizes for populations 1 and 2. Assume that 01 21 and 02 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 ni 15, and the sample mean X1 1.62. For the sample drawn from mortgages that were not refinanced, the sample size n2 00, and the sample mean X2 1.23. (Note: The sample means match those from the study, but the sample sizes have been reduced.) The point estimate of H1 H2 is

Explanation / Answer

n1 = 115 , n2 = 100 , s1 = 1.21 , s2 = 0.95 , x1 = 1.62 , x2 = 1.23

z value at 95% CI = 1.96

CI = ( x1 - x2 ) +/- z * sqrt ( s1^2 / n1 + s2^2 / n2)
= ( 1.62- 1.23) + /- 1.96 * sqrt ( 1.21^2 / 115 + 0.95^2 / 100)
= (0.101 , 0.679)

Hypothesis:

Ho : mu1 - mu2 = 0 , H1 : mu1 - mu2 not equal to 0

Test statistic:

SE = sqrt[(s1^2/n1) + (s2^2/n2)]
SE = sqrt[(1.21^2/115) + (0.95^2/100)]
= 0.1475

z = [ (x1 - x2) - d ] / SE
= [ (1.62 - 1.23) - 0 ] / 0.1475
= 2.644

p value is calculated using z = 2.644
P value = 0.00819

A level of significance of alpha =0.05 is specified for the study . The null hypothesis is rejected

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