AMS$578$Home work$2$1. Refer to Grade point average problem in homework 1 a. Prepare a box plot for the ACT scores Xi. Are there any noteworthy features in this plot? b. Prepare a dot plot of the residuals. What information does this plot provide? c. Plot the residual ei against the fitted values Ŷi. What departures from regression model (2.1) can be studied from this plot? What are your findings? d. Prepare a normal probability plot of the residuals. Also obtain the coefficient of correlation between the ordered residuals and their expected values under normality. Test the reasonableness of the normality assumption here using Table B.6 and α =.05. What do you conclude? e. Conduct the Brown- Forsythe test to determine whether or not the error variance varies with the level of X. Divide the data into the two groups, X < 26, X >= 26, and use α = .01. State the decision rule and conclusion. Does your conclusion support your preliminary findings in part (c)? f. Information is given below for each student on two variables not included in the model, namely, intelligence test score (X2) and high school class rank percentile (X3). (Note that larger class rank percentiles indicate higher standing in the class, e.g., 1% is near the bottom of the class and 99% is near the top of the class.) Plot the residuals against X2 and X3 on the separate graphs to ascertain the model can be improved by including either of these variables. What do you conclude? 2. Refer to Grade point average problem. Assume that linear regression through the origin model (4.10) is appropriate. a. Fit regression model (4.10) and state the estimated regression function. b. Estimate β1 with a 95 percent confidence interval . Interpret your interval estimate. c. Estimate the mean freshman GPA for students whose ACT test score is 30. Use a 95 percent confidence interval. 3. Refer to Grade point average problem. a. Plot the fitted regression line and the data. Dose the linear regression function through the origin appear to be a good fit here? b. Obtain the residuals ei. Do they sum to zero? Plot the residual against the fitted values. What conclusions can be drawn from your plot? c. Conduct a formal test for lack of fit of linear regression through the origin; use α = 0.005. State the alternatives, decision rule, and conclusion. What is the P-value of the test? i: 1 2 3 … 118 119 120 X2: 122 132 119 … 140 111 110 X3: 99 71 75 … 97 65 854. Refer to Copier maintenance problem in homework 1 a. Prepare a dot plot for the number of copiers serviced Xi. What information is provided by this plot? Are there any outlying cases with respect to this variable? b. The cases are given in time order. Prepare a time plot for the number of copiers serviced. What does your plot show? c. Prepare a stem-and- leaf plot of the residuals. Are there any noteworthy features in this plot? d. Prepare residual plots of ei versus Ŷi and ei versus Xi on separate graphs. Do these plots provide the same information? What departures from regression model (2.1) can be studied from these plots? State your findings. e. Prepare a normal probability plot of the residuals. Also obtain the coefficient of correlation between the ordered residual s and their expected values under normality. Does the normality assumption appear to be tenable here? Use Table B.6 and α = .10. f. Prepare a time plot of the residuals to ascertain whether the error terms are correlated over time. What is your conclusion? g. Assume that (3.10) is applicable and conduct the Breusch-Pagan test to determine whether or not the error variance varies with the level of X. Use α = .05. State the alternatives decision rule, and conclusion. h. Information is given below on two variables not included in the regression model, namely, mean operational age of copiers serviced on the call (X2, in months) and year of experience of the service person making the call (X3). Plot the residuals against X2 and X3 on separate graphs to ascertain whether the model can be improved by including either or both of these variables. What do you conclude? 5. Refer to Copier Maintenance problem. a. Will b0 and b1 tend to err in the same direction or in opposite directions here? Explain. b. Obtain Bonferroni joint confidence intervals for β0 and β1, using a 95 percent family confidence coefficient. c. A consultant has suggested that β0 should be and β1 should equal 14.0. Do your joint confidence intervals in part (b) support this view? 6. Refer to Copier Maintenance problem. a. Estimate the expected number of minutes spent when there are 3, 5, and 7 copiers to be serviced, respectively. Use interval estimates with a 90 percent family confidence coefficient based on the Working-Hotelling procedure. b. Two service calls for preventive maintenance are scheduled in which the numbers of copiers to be serviced are 4 and7, respectively. A family of prediction intervals for the time to be i: 1 2 3 … 43 44 45 X2: 20 19 27 … 28 26 33 X3: 4 5 4 … 3 3 6spent on these calls is desired with a 90 percent family confidence coefficient. Which procedure, Scheffè or Bonferroni, will provide tighter prediction limits here? c. Obtain the family of prediction intervals required in part (b), using the more efficient