Department of Statistics The Wharton School University of Pennsylvania Statistics 621 Fall 2008 Solutions, Final Exam (1) (2) Assuming the MRM holds, if the 95% confidence interval for the intercept β 0 in a multiple regression model is the interval [-18, 54], then the standard error of the estimated intercept is about 18. The length of the 95% confidence interval is approximately 4 times the SE. Since the interval includes zero, the estimate is not statistically significantly different from zero. (3) The intercept is often an extrapolation in a regression model because zero lies outside the range of the explanatory variables. An extrapolation occurs when the value used for the explanatory variable lies outside the range of the data. This often occurs for the intercept since the intercept is the fitted value when x = 0. (4) A narrow cluster of observations at the center of the leverage plot of an explanatory variable in a multiple regression suggests that this variable is collinear with other explanatory variables . The position of points on the horizontal axis of the leverage plot is determined by the variation in the variable that remains after removing the effects of other explanatory variables. In the presence of high correlation among the X’s, there’s little such variation left. (5) In order to build a regression model that estimates a constant marginal elasticity of sales with respect to price, we must fit a simple regression of log sales on log price . The slope in a log-log model is the elasticity. (6) The explanatory variables in a multiple regression are assumed by the multiple regression model to be linearly related to the response . Normality is only assumed about the error variation. The values of the explanatory variable are set by the experimenter and not part of the assumptions of the model. (7) The overall F -ratio in a multiple regression with K explanatory variables (found in the analysis of variance table ) tests H 0 : β 1 = β 2 = … = β K = 0 . This is the overall F-test of the entire model, measuring whether R 2 is larger than could be expected from unrelated variables. (8) To check for the presence of heteroscedasticity in a multiple regression model, it is recommended that we plot the residuals of the model versus the fitted values . Heteroscedasticity often occurs as the predicted values get larger; this plot shows the data in order of increasing predicted values.
Statistics 621, Solutions to the Final Exam -2- Q1, 2008 (9) In order to allow the effect of the explanatory variable Advertising (which measures promotional spending) on the response Sales (which measures retail sales at various stores) to depend upon the geographic Location of the stores (a categorical variable), we should include Location and its interaction with Advertising in our model .
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