Applied Regression Analysis

1 EPIB 651: Applied Regression Analysis Fall 2022 Homework 2 (Due: October 06, 2022, 12:30PM) Problem 1 Following a recent congressional election, a political scientist attempted to investigate the relationship between campaign expenditures on television advertisements and subsequent voter turnout. The following table presents the percentage of total campaign expenditures delegated to television advertisements (TVEXP) and the percentage of registered voter turnout (VOTE) for a hypothetical sample of 20 congressional districts. a. Determine the least-squares estimates of the slope and intercept for the straight-line regression of VOTE (Y) on TVEXP (X). Draw the estimated line on a scatter diagram, and comment on the fit. b. Test the null hypothesis that the true slope is 0; be sure to interpret your result. c. Obtain a 95% confidence interval for β1. Interpret your result. d. Would you reject the null hypothesis H0: β1 = 0 based on the confidence interval you calculated in part (c)? Explain. e. Determine and sketch 95% confidence bands on the scatter diagram you generated in (a). Use your diagram to estimate the mean VOTE when TVEXP = 50. Interpret your result. 2 Problem 2 Sales revenue (Y) and advertising expenditure (X) data for a large retailer for the period 1988– 1993 are given in the following table. a. Based on the accompanying scatter diagram of Sales revenue versus Advertising expenditure, does there appear to be a linear relationship between these two variables? b. State the model for the straight-line regression of revenue (Y) on Advertising expenditure (X). Determine the least-squares estimates for this regression line. Interpret the estimated values of the slope and the intercept in the context of the problem. c. Test for the significance of the slope parameter of the model in part (b). Interpret your result. d. Determine a 95% confidence interval for the true slope in part (c). Interpret your result with regard to the test mentioned in part (c). 3 Problem 3 A panel of educators in a large urban community wanted to evaluate the effects of educational resources on student performance. They examined the relationship between twelfth grade mean verbal SAT scores (Y) and the following independent variables for a random sample of 25 high schools: X1 = Per pupil expenditure (in dollars); X2 = Percentage of teachers with a master’s degree or higher; and X3 = Pupil–teacher ratio. The sums of squares shown next can be used to summarize the key results from the regression of Y on X1, X2, and X3: SSY = 28,222.23 SSE = 2,248.23 a. Determine the ANOVA table for the regression of Y on X1, X2, and X3. b. Determine the R2-value for the model in part (a). Based on this value, comment on whether the three educational resource variables appear to be associated with student performance. Problem 4 A psychiatrist wants to know whether the level of pathology (Y) in psychotic patients 6 months after treatment can be predicted with reasonable accuracy from knowledge of pretreatment symptom ratings of thinking disturbance (X1) and hostile suspiciousness (X2). Data were collected on 53 patients. Answer the following questions using this dataset. The CSV data file "HW2P4.csv" can be downloaded from CANVAS. a. Conduct overall regression F tests for these three models: Y regressed on X1 and X2; Y regressed on X1 alone; and Y regressed on X2 alone. Based on your answers, how would you rate the importance of the two variables in predicting Y? b. Perform variables-added-in-order tests for both variables, with X1 added first. Use 0.05 = . c. Perform variables-added-in-order tests for both variables, with X2 added first. Use 0.05 = . d. Provide a table of variables-added-last tests for both X1 and X2. e. Based on your results in parts (a) to (d), what is the most appropriate model to use? 4 Problem 5 An educator examined the relationship between the number of hours devoted to reading each week (Y) and the independent variables social class (X1), number of years of school completed (X2), and reading speed (X3), in pages read per hour. The following ANOVA table was obtained from a stepwise regression analysis on data for a sample of 19 women over 60. Source d.f. SS (X3) 1 1,058.628 Regression (X2|X3) 1 183.743 (X1|X2, X3) 1 37.982 Residual 15 363.300 a. Provide a test to compare the following two models: 0 1 1 2 2 3 3Y X X X E   = + + + + and 0 3 3Y X E = + + b. Provide a test to compare the following two models: 0 2 2 3 3Y X X E  = + + + and 0Y E= + . c. State which two models are being compared in the computation: 37.982 363.3 /15 F = .