This problem will use the subprime data to walk you through calculating a difference in means test. To begin this problem first download subprime data and load it into STATA. These are data collected by the U.S. government on all home-lending transactions in Cape Coral and Fort Myers (Florida). They contain information on each loan applicant and give information on whether that applicant received a subprime loan (highrate) as well as on the amount of the loan in $10,000 units (loanamount). They also contain basic demographic information such as race, gender, and income. For the remainder of the problem, we will treat this dataset as the entire, true population. Unless otherwise stated, you can assume that the samples we will draw are large enough to use the Normal approximation.
You are a policy researcher trying to unpack what happened in the recent U.S. foreclosure crisis. You have narrowed your research to the Cape Coral-Fort Myers area, the area of the United States most devastated by the foreclosure crisis. Suppose a lawsuit has been _led in U.S. District Court by a group of Fort Myers women who claim that women in the area were loaned less money than men. The defendants- a group of local mortgage lenders- are vigorously denying these claims, and the case is now advancing to trial. Having heard about your expertise in this area, the federal judge hearing the case has brought you in to provide expert testimony. Your task in this problem is to assist the judge in her determination.
Using STATA, calculate (1) the average loan amount for women in your sample, (2) the average loan amount for men, and (3) and (4), the sample standard deviation for each. Report those results in a nicely formatted table.
Let μ_m and μ_w equal the average loan amount for men and women. Logically, then, the defendants are arguing that μ_m-μ_w = 0, while the plaintiffs believe that μ_m-μ_w > 0. Treat the defendant's argument as the null hypothesis, and derive the test statistic for this hypothesis. Do not use STATA for this part and be sure to show all your work.
The judge wants to know whether to (1) dismiss the case or to (2) allow the case to proceed. She would like to dismiss if the defendants are right, and she would like the case to proceed if the women's group is right. Given this information, explain what the Type I and the Type II errors are for this problem.
What are the rejection regions of this test statistic for a = 0.05 and a= 0.01? Does the test statistic for your sample fall in these regions? Can you reject the null hypothesis?
E) Fortunately, STATA has commands so you don't have to compute test statistics by hand. Use the following command to conduct the test you just completed:
ttest loanamount, by(woman) unequal
The last argument to this command, unequal tells STATA not to assume equal variances for the two groups. Report the 95% confidence interval for each mean. Does the confidence interval for the sample mean of women include the sample mean for men? Does the confidence interval for the difference include zero? What is the p-value for the test statistic? Is it less than a? What does this suggest to you about confidence intervals of means, the confidence interval of the difference in means, and hypothesis tests about the difference?
F) Discuss the conclusions of your research, giving the judge your ultimate recommendation. Justify your recommendation using language that non-POL 551 students could understand. Should she dismiss the case? Or do the women's rights groups have a basis to proceed?