2 a) Is the estimated multiple linear regression model of Sleep with all variables listed in Problem 1 statistically significant at = 001? (State the relevant hypothesis test, rejection rule and your conclusion) b) Compute 2 of the multiple linear regression model and interpret it. c) Which two predictors are most highly correlated with each other? Which two explanatory variables exhibit the smallest correlation? d) Fill in an empty cell (Sleep/Female) in the correlation matrix. e) Consider simple regressions of Sleep on each explanatory variable. Identify predictor(s) that are statistically significant at = 005 in the simple regressions but are not statistically significant at = 005 in the multiple regression. Give a reason for their lack of statistical significance in the multiple regression. f ) Before fitting the multiple regression models the students hypothesized that for every one cups increase in CoeeCups, the expected decrease in Sleep is 0.4 of an hour. Based on the estimated model were they correct? Answer by constructing an appropriate 95% confidence interval or by using a hypothesis test with = 005 g) They also thought that the average sleep time in the population of NYU undergraduate student is more that 7 hours. Is there evidence in the data, at = 005 that they were right? (State the hypothesis test, rejection rule and your conclusion) 2 Minitab output for problems 1 and 2 The regression equation is Sleep = 6.62 - 0.708 Female Predictor Constant Female Coef 6.6250 -0.7083 S = 1.07802 SE Coef 0.2201 0.3112 R-Sq = 10.1% T 30.11 P 0.000 R-Sq(adj) = 8.2% Descriptive Statistics: Sleep Variable Sleep N 48 N* 0 Mean 6.271 SE Mean 0.162 StDev 1.125 Minimum 4.000 Q1 5.250 Median 6.000 Q3 7.000 The regression equation is Sleep = 5.94 + 0.0173 Age + 0.227 Study - 0.122 Extra - 0.583 Female - 0.301 Cupscoffee Predictor Constant Age Study Extra Female Cupscoffee Coef 5.9399 0.01734 0.22737 -0.1215 -0.5826 -0.3009 SE Coef 0.9384 0.04436 0.07882 0.1231 0.2757 0.1060 T 6.33 0.39 2.88 -0.99 -2.11 -2.84 P 0.000 0.698 0.006 0.329 0.041 0.007 MS 4.7874 0.8462 F 5.66 S = 0.919914 Analysis of Variance Source Regression Residual Error Total DF 5 42 47 SS 59.4792 P The regression equation is Sleep = 5.98 + 0.227 Study - 0.377 Cupscoffee Predictor Constant Study Cupscoffee S = 0.957993 Coef 5.9755 0.22720 -0.3765 SE Coef 0.3433 0.07493 0.1015 R-Sq = 30.6% T 17.41 3.03 -3.71 P 0.000 0.004 0.001 R-Sq(adj) = 27.5% Maximum 9.000 Correlations: Sleep, Age, Study, Extra, Cupscoffee, Female Sleep 0.256 0.079 Age Study 0.306 0.035 0.373 0.009 Extra -0.290 0.045 -0.167 0.256 0.090 0.542 Cupscoffee -0.405 0.004 0.004 0.977 0.163 0.268 0.387 0.007 -0.155 0.293 0.089 0.547 0.156 0.288 Age Female Cell Contents: Pearson correlation P-Value Study Extra Cupscoffee 0.166 0.259