AMS 578 Homework 3 1. Brand Preference. In a Small-scale experiment study of the relation between degree of brand liking (Y) and moisture content (X1) and sweetness(X2) of the product, the following results were obtained from the experiment based on a completely randomized design (data are coded): (The dataset can be downloaded from: https://netfiles.umn.edu/users/nacht001/www/nachtsheim/Kutner/Chapter%20%206%20Data%20Sets/CH06PR05.txt) a. Fit regression model (6.1) to the data. State the estimated regression function. How is b1 interpreted here? b. Plot the residuals against Ŷ, X1, X2 and X1X2 on separate graphs. Also prepare a normal probability plot. Interpret the plots and summarize your findings. c. Test whether there is a regression relation, using α =.01. State the alternation, decision rule, and conclusion. What does your test imply about β1 and β2? d. What is the P-value of the test in part (a)? e. Calculate the coefficient of multiple determination R2. How is interpreted here? i: 1 2 3 … 14 15 16 Xi1: 4 4 4 … 10 10 10 Xi2: 2 4 2 … 4 2 4 Yi: 64 73 61 … 95 94 100 2. Refer to Brand Preference problem. Assume that regression model (6.1) with independent normal error terms is appropriate. a. Obtain an interval estimate of E(Yh) when Xh1=5 and Xh2=4. Use a 99 percent confidence coefficient. Interpret your interval estimate. b. Obtain a prediction interval for a new observation Yh(new) when Xh1=5 and Xh2=4. Use a 99 percent confidence coefficient. c. Obtain the analysis of variance table that decomposes the regression sum of squares into extra sums of squares associated with X1 and with X2, given X1. d. Test whether with X2 can be dropped from the regression model given that with X1 is retained, Use the F* test statistic and level of significance .01. State the alternatives, decision rule, and conclusion. What is the P-value of the test? 3. Grocery retailer. A large, national grocery retailer tracks productivity and costs of its facilities closely. Data below were obtained from a single distribution center for a one-year period, each data point for each variable represents one week of activity. The variables included are the number of cases shipped (X1), the indirect costs of the total labor hours as a percentage (X2), a qualitative predictor called holiday that is coded 1 if the week has a holiday and 0 otherwise (X3), and the total labor hours (Y). (The dataset can be downloaded from: https://netfiles.umn.edu/users/nacht001/www/nachtsheim/Kutner/Chapter%20%206%20Data%20Sets/CH06PR09.txt)a. Fit regression model (6.5) to the data for three predictor variables. State the estimated regression function. How are b1, b2, and b3 interpreted here? b. Plot the residuals against Ŷ, X1, X2, X3,and X1X2 on separate graphs. Also prepare a normal probability plot. Interpret the plots and summarize your findings. c. Prepare a time plot of the residuals. Is there any indication that the error terms are correlated? Discuss. Xi1: i: 1 2 3 … 50 51 52 305,657 328,476 317,164 … 290,455 411,750 292,087 Xi2: 7.17 6.20 4.61 … 7.99 7.83 7.77 Xi3: 0 0 0 … 0 0 0 Yi: 4264 4496 4317 … 4499 4186 4342 4. Refer to Grocery retailer problem. a. Obtain the analysis of variance table that decomposes the regression sum of squares into extra sums of squares associated with X1; with X3, given X1; and with X2, given X3 and X1? b. Test whether X2 can be dropped from the regression model given that X1 and X3 are retained? Use the F* test statistic and α =.05. State the alternatives, decision rule, and conclusion. What is the P-value of the test? c. Does SSR(X1) +SSR(X2| X1) equal SSR(X2) +SSR(X1| X2) here? Must this always be the