CRPLAN 6500 Homework 4 Due: Feb. 18 (on Thursday) Name: Last Name.#: Please round off the final answers to two decimal places. But, in the process of calculation, it is recommended to round off to...

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CRPLAN 6500 Homework 4 Due: Feb. 18 (on Thursday) Name: Last Name.#: Please round off the final answers to two decimal places. But, in the process of calculation, it is recommended to round off to three or four decimal places. 1. (20pts) Based on the 2018 city records, the average number of household vehicles was reported as 3, with a standard deviation of 1. An investigator claims that recent changes in car ownership patterns might have changed this number. In order to support his argument, the investigator randomly selects 100 households and finds that the sample mean is 2.8 vehicles per household. Does the investigator have enough evidence to claim household vehicle ownership decreased significantly? Conduct a one-sided test at 5% significant level. What is your conclusion? 2. To investigate possible gender discrimination in a firm, a sample of 100 men and 64 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries follows. Average Salary Standard Deviation N Men $3150 $200 100 Women $3000 $320 64 a) (15pts) What do these data suggest about wage differences in the firm? Do they represent statistically significant evidence that average wages of men and women are different? Use a two-sided test at 5% level. Compute the relevant t-statistic. b) (15pts) Compute the p-value associated with the t-statistics. Based on the p-value you calculated, is there statistically significant evidence that average wages of men are different than that of women? 3. Grades on a standard test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected students in Florida; in this sample, the mean is 1050 and the standard deviation is 108. a) (15pts) Is there statistically significant evidence that the Florida students perform differently than other students in the United States? (Use a two-sided test at 5% significant level) b) (15pts) Another sample of 503 students are selected at random from Florida. They are given a three-hour preparation course before the test is administered. Their average test score is 1055 with a standard deviation of 95. Is there statistically significant evidence that the prep course helped? (Use a one-sided test at 5% significance level to test the two samples from Florida) 4. U.S. News and World Report ranks colleges and universities annually. You randomly sample 100 of the national universities and liberal arts colleges from the year 2019 issue. The average cost, which includes tuition, fees, and room and board, is $23,570 with a standard deviation of $7,015. a) (10pts) Based on this sample, construct a 90% confidence interval of the average cost of attending a university/college in the United States. Cost varies quite a bit. One of the reasons may be that some universities/colleges have a better reputation than others. U.S. News and World Report try to measure this factor by asking university presidents and chief academic offers about the reputation of institutions. The ranking is from 1 (“marginal”) to 5 (“distinguished”). You decide to split the sample according to whether the academic institution has a reputation of greater than 3.25 or not. This gives you the statistics shown in the table below. Reputation Category Average Cost () Std. Dev. of Cost () N Ranking 3.25 $29,311.31 $5,649.21 30 Ranking 3.25 $21,227.06 $6,133.38 70 b) (10pts) Test the hypothesis that the average cost for all universities/colleges is the same, independent of the reputation. Conduct a two-sided test at the 5% level. What is your conclusion? 2