Cоnsider а study оf sаles figures fоr n = 33 firms аlong with their customer rating scores. It is apparent that Y = Sales has a positive association with X = Rating. An appropriate regression model relating Sales to Rating could be useful for predicting Sales based on Rating. The most straightforward approach would be to fit a simple linear regression (SLR) model for Y vs X, provided that the LINE assumptions are satisfied. Type your answers to the following questions in the text box below making sure to reference the relevant Minitab output in your answers. a. (7 pts) Residual plots for an SLR model for Y vs X are as follows.Use the plots to determine if the LINE assumptions are satisfied, making sure to include a numerical test when checking for normality. b. (7 pts) Your analysis in part (a) should have indicated natural log transformations could be usefully applied to both X and Y. Residual plots for an SLR model for ln(Y) vs ln(X) are as follows. Use the plots to determine if the LINE assumptions are better satisfied for this model relative to the model in part (a), making sure to include a numerical test when checking for normality. c. (5 pts) Use relevant parts of the following output based on the model in part (b) to compute a 95% confidence interval for the mean Sales expected for firms with a rating of 10 based on the fitted model in part (c). [Hint: Not all the output is relevant. Remember to take into account the transformations to X and Y.] d. (5 pts) Use relevant parts of the output from part (c) to compute a 95% prediction interval for the Sales predicted for an individual firm with a rating of 10 based on the fitted model in part (c). [Hint: Not all the output is relevant. Remember to take into account the transformations to X and Y.]
Cоnsider а study оf Mаth scоres (Mаth) for n=14 students with information about their undergraduate major (Major) and weekly hours of study (Hours). The goal was to fit a regression model to express the dependence of Y (Math) on X (Hours) and Major (Engineering, History, or Science). Type your answers to the following questions in the text box below making sure to reference the relevant Minitab output in your answers. a. (6 pts) Clearly define a set of indicator variables that could be used in a regression model to represent the qualitative variable Major. [Hint: Think carefully about the number of indicator variables needed given the number of levels of Major and use "Engineering" as the reference level.] b. (6 pts) Write a population multiple linear regression equation for predicting the Math scores in terms of Hours and Major. Since Major could impact the dependence of Math score (Y) on Hours (X), include in the model interaction effects between Hours and Major, together with their main effects. [Hint: Your equation should include Y, X, the indicator variables you defined in part (a), interaction terms, and population regression coefficients (β’s). Do not include estimated coefficients, i.e., numbers, in this part.] c. (8 pts) Conduct a single hypothesis test based on the model from part (b) to determine whether the average change in Math score per one additional Hour of study per week differs by Major. Write the null and alternative hypotheses, the test statistic, the p-value, and the conclusion based on a significance level of 0.05. Use relevant parts of the following Minitab output to support your answer. [Not all the output is relevant.] d. (8 pts) Write a new population regression model based on your conclusion to part (c). Then conduct two separate hypothesis tests based on this new model for whether the mean Math score for a fixed number of Hours of study per week differs by Major. For each test, write the null and alternative hypotheses, the test statistic, the p-value, and the conclusion based on a significance level of 0.05. Use relevant parts of the following Minitab output to support your answer. [Not all the output is relevant.] e. (6 pts) Based on your conclusion to part (d), write three fitted regression equations that can be used to predict the Math score from the Hours of study per week for each Major. [Hint: Your equations should include number values, not β’s.] f. (4 pts) Based on one of the equations from part (e), predict the Math score for a History major who studies 4 hours a week. A point estimate is sufficient.