AMS578Homework1(DueFeb23,2009)PleaseemailquestionsonHWtoourTA,ShaonanZhang[email protected]TAOfficeHour:Wed1:30‐3:30pm1. Grade point average. The director of admissions of a small college selected 120 students at random from the new freshman class in a study to determine whether a student’s grade point average(GPA) at the end of the freshman year(Y) can be predicted from the ACT test score (X). The results of the study follow. Assume that first-order regression model(1.1) is appropriate. (The dataset can be downloaded from https://netfiles.umn.edu/users/nacht001/www/nachtsheim/Kutner/Chapter%20%201%20Data%20Sets/CH01PR19.txt ) i: 1 2 3 … 118 119 120 Xi: 21 14 28 … 28 16 28 Yi: 3.897 3.885 3.778 … 3.914 1.860 2.948 a. Obtain the least squares estimates of β0 and β1, and state the +estimated regression function. b. Plot the estimated regression function and the data. Does the estimated regression function appear to fit the data well? c. Obtain a point estimate of the mean freshman GPA for students with ACT test score X= 30. d. What is the point estimate of the change in the mean response when the entrance test score increases by one point? 2. Refer to Grade point average problem. a. Obtain a 99 percent confidence interval for β1. Interpret your confidence interval. Does it include zero? Why might the director of admissions be interested in whether the confidence interval includes zero? b. Test, using the test statistic t*, whether or not a linear association exists between student’s ACT score(X) and GPA at the end of the freshman year(Y). Use a level of significance of .01. State the alternatives, decision rule and conclusion. c. What is the P-value of your test in part (f)? How does it support the conclusion reached in part (f)?3. Refer to Grade point average problem. a. Obtain a 95 percent interval estimate of the mean freshman GPA for students whose ACT test score in 28. Interpret your confidence interval. b. Mary Jones obtained a score of 28 on the entrance test. Predict her freshman GPA using a 95 percent prediction interval. Interpret your prediction interval. c. Is the prediction interval in part (b) wider than the confidence interval in part (a)? Should it be? d. Determine the boundary values of the 95 percent confidence band for the regression line when Xh =28. Is your confidence band wider at this point than the confidence interval in part (a)? Should it be? 4. Copier maintenance. The Tri-City Office Equipment Corporation sells an imported copier on a franchise basis and performs preventive maintenance and repair service on this copier. The data below have been collected from 45 recent calls on users to perform routine preventive number of minutes spent by the service person. Assume that first-order regression model (1.1) is appropriate. (The dataset can be downloaded from https://netfiles.umn.edu/users/nacht001/www/nachtsheim/Kutner/Chapter%20%201%20Data%20Sets/CH01PR20.txt ) i: 1 2 3 … 43 44 45 Xi: 2 4 3 … 2 4 5 Yi: 20 60 46 … 27 61 77 a. Obtain the estimated regression function. b. Plot the estimated regression function and the data. How well does the estimated regression function fit the data? c. Interpret b0 in your estimated regression function. Does b0 provide any relevant information here? Explain. d. Obtain a point estimate of the mean service time when X = 5 copiers are serviced. 5. Refer to Copier Maintenance problem. a. Estimate the change in the mean service time when the number of copiers serviced increases by one. Use a 90 percent confidence interval. Interpret your confidence interval.b. Conduct a t test to determine whether or not there is a linear association between X and Y here; control the α risk at .10. State the alternatives, decision rule, and conclusion. What is the P-value of your test? c. Are your results in parts (e) and (f) consistent? Explain. d. The manufacturer has suggested that the mean required time should not increase by more than 14 minutes for each additional copier that is serviced on a service call. Conduct a test to decide whether this standard is being satisfied by Tri-City. Control the risk of a Type I error at .05. State the alternatives, decision rule, and conclusion. What is the P-value of the test? e. Does b0 give any relevant information here about the “start-up” time on calls –i.e., about the time required before service work is begun on the copiers at a customer location? 6. Refer to Copier Maintenance problem. a. Obtain a 90 percent confidence interval for the mean service time on calls in which sic copier are serviced. Interpret your confidence interval. b. Obtain a 90 percent prediction interval for the service time one the next call in which six copiers are serviced. Is your prediction interval wider than the corresponding confidence interval in part (a)? Should it be? c. Management wishes to estimate the expected service time per copier on calls in which six copiers are serviced. Obtain an appropriated 90 percent confidence interval by converting the interval obtained in part (a). Interpret the converted confidence interval. d. Determine the boundary values of the 90 percent confidence band for the regression line when Xh = 6. Is your confidence band wider at this point than the confidence interval in part (a)? Should it