STAT-UB.0103.04-Statistics for Business Control and Regression Models December 9, 2015 Homework 8-due December 15 Reading: From MBS: 12.1-12.4, 12.6 (up to page 700), 12.7, 12.11-12.12 Lecture Notes #22 1) A survey of a random sample of 47 NYU undergraduate students was conducted to investigate the relationship between Sleep=average sleep time per night (in hours) and the following variables: Study =average study time per day (in hours) CupsCoee =average number of cups of coee consumed by a student per day, and Age (in years) Answer questions below based on the Minitab output on the next two pages. a) Is the estimated multiple linear regression model of Sleep on the variables listed above statistically signicant at = 001? (State the relevant hypothesis test and your conclusion) b) Compute 2 of the multiple linear regression model and interpret it. c) Which two explanatory variables are most highly correlated with each other? Which two explanatory variables exhibit the smallest correlation? d) Identify predictor(s) that are statistically signicant at = 005 in the simple regressions but are not statistically signicant at = 005 in the multiple regression. Give a reason for their lack of statistical signicance in the multiple regression. e) The two-variable model with CupsCoee and Study was chosen as the nal model. Interpret the coecients in this model. f ) Before tting the multiple regression models the students hypothesized that for every one cup increase in CupsCoee, the expected decrease in Sleep is 0.4 of an hour. Based on the nal model were they correct? Answer by constructing an appropriate 95% condence interval. g) Construct a 95% condence interval for the average sleep time of students who study 4 hours and drink 6 cups of coee. Is this a valid condence interval? Explain. 1 Regression Analysis: Sleep versus Age, Study, Cupscoffee The regression equation is Sleep = 4.91 + 0.0449 Age + 0.231 Study - 0.357 Cupscoffee Predictor Constant Age Study Cupscoffee Coef 4.9100 0.04486 0.23083 -0.35740 SE Coef 0.8070 0.03955 0.07268 0.09061 T 6.08 1.13 3.18 -3.94 P 0.000 0.263 0.003 0.000 S = 0.852758 Analysis of Variance Source Regression Residual Error Total DF 3 43 46 SS 20.6029 31.2694 51.8723 MS 6.8676 0.7272 F 9.44 P 0.000 Correlations: Sleep, Age, Study, Cupscoffee Sleep 0.296 0.043 Cupscoffee 0.369 0.011 -0.410 0.004 Study Age 0.390 0.007 Age 0.001 0.997 Study 0.155 0.297 Regression Analysis: Sleep versus Study, Cupscoffee The regression equation is Sleep = 5.75 + 0.262 Study - 0.364 Cupscoffee Predictor Constant Study Cupscoffee Coef 5.7544 0.26166 -0.36377 S = 0.855534 SE Coef 0.3130 0.06763 0.09073 R-Sq = 37.9% T 18.39 3.87 -4.01 P 0.000 0.000 0.000 R-Sq(adj) = 35.1% Analysis of Variance Source Regression Residual Error Total DF 2 44 46 SS 19.6671 32.2053 51.8723 MS 9.8335 0.7319 F 13.43 P 0.000 Predicted Values for New Observations New Obs 1 Fit SE Fit 0.417 Values of Predictors for New Observations New Obs 1 Study 4.00 Cupscoffee 6.00 Descriptive Statistics: Sleep, Study, Cupscoffee Variable Sleep Study Cupscoffee N 47 47 47 N* 0 0 0 Variable Sleep Study Cupscoffee Maximum 9.000 10.000 8.000 Mean 6.213 4.000 1.617 SE Mean 0.155 0.275 0.205 StDev 1.062 1.888 1.407 Minimum 4.000 1.000 0.000 Q1 5.000 3.000 1.000 Median 6.000 4.000 1.000 Q3 7.000 5.000 2.000 2) In the undergraduate statistics class at Stern there were students from Stern School of Business, Tisch School of Arts and from Steinhardt. Consider those students as a random sample from the population of NYU students who take statistics classes at Stern. A professor who taught the class was interested in the relationship between the midterm exam grade (recorded on the scale from 0 to 40) and the homework grade (expressed as percentage of perfect homework grade), and the type of undergraduate college, which was coded as follows: Tisch = 1 if a student was from Tisch = 0 if not and Steinhardt = 1 if a student was from Steinhardt 0 if not Answer the following questions with the help of Minitab output on the next page. a) Interpret the coecients in the estimated model. b) Do Tisch students, who have the same HW percentage as Stern students, have on average the same midterm grade as Stern students? Test at = 005 (State and interpret the hypothesis test, obtain rejection rule and state your conclusion.) c) Do Steinhardt students who have the same HW percentage as Stern student have the same average midterm grade as Stern students? (To answer this question construct an appropriate 95% condence interval) c) One of the students in the class got a perfect midterm score of 40. He was a Stern student whose homework percentage was 90%. What is the residual associated with this observation? d) An alternative model for the relationship of midterm grade to Homework percentage and school type was proposed in which type of school was coded with the predictor School which took on three values: 1, 2, and 3, where 1=Tisch, 2=Steinhardt, 3=Stern. Interpret the coecient of School. Discuss the dierence between the two estimated models, the one in which School is coded with two dummy variables and the one with School taking vaues 1,2,3. Which model do you prefer and why? 2 Regression Analysis: Midterm Grade versus HW Percent, Tisch, Steinhardt The regression equation is Midterm Grade = 19.3 + 0.181 HW Percent - 6.43 Tisch - 5.68 Steinhardt Predictor Constant HW Percent Tisch Steinhardt Coef 19.254 0.18060 -6.430 -5.681 S = 5.81915 SE Coef 4.318 0.04990 3.325 1.871 R-Sq = 42.6% T 4.46 3.62 P 0.000 0.001 -3.04 0.004 R-Sq(adj) = 38.2% Analysis of Variance Source Regression Residual Error Total DF 3 39 42 SS 979.48 1320.64 2300.12 MS 326.49 33.86 F 9.64 P 0.000 Regression Analysis: Midterm Grade versus HW Percent, School The regression equation is Midterm Grade = 6.99 + 0.169 HW Percent + 4.25 School Predictor Constant HW Percent School Coef 6.99 0.169 4.25 S = 5.84314 SE Coef 4.635 0.049 1.405 R-Sq = 40.6% T 1.51 3.45 3.02 P 0.139 0.001 0.004 R-Sq(adj) = 37.7% Analysis of Variance Source Regression Residual Error Total DF 2 40 42 SS 934.43 1365.69 2300.12 MS 467.21 34.14 F 13.68 P 0.000 3) Consider the regression of obesity rate (in percent) across the 50 states of US on the following variables: MI=Median personal income (in dollars) LE=Life expectancy at birth (in years) EI=Education index=Percentage of the adult population that has completed a high school diploma or its equivalent, a four-year bachelor's degree, or a graduate degree. HDI ranking= composite measure made up of three other indices that measure health, education, and income Answer questions below based on the minitab output that follows to on the next page a) Is this a statistically signicant regression at = 001? b) Are the coecient of the individual variables statistically signicant in the multiple regression at = 01? c) Is there anything puzzling about the results of this regression? Explain. 3 Regression Analysis: Obesity versus MI, LE, EI, HDI Ranking The regression equation is Obesity = 178 - 0.00034 MI - 1.92 LE - 0.20 EI + 2.4 HDI Ranking Predictor Constant MI LE EI HDI Ranking Coef 178.3 -0.000341 -1.922 -0.200 2.36 S = 1.95890 SE Coef 153.5 0.001160 2.174 4.839 14.89 R-Sq = 67.6% T 1.16 -0.29 -0.88 -0.04 0.16 P 0.252 0.770 0.381 0.967 0.875 R-Sq(adj) = 64.7% Analysis of Variance Source Regression Residual Error Total DF 4 45 49 SS 359.874 172.679 532.552 MS 89.968 3.837 F 23.45 P 0.000 Correlations: Obesity, MI, LE, HDI Ranking, EI Obesity -0.522 0.000 MI LE -0.813 0.000 0.527 0.000 HDI Ranking -0.742 0.000 0.867 0.000 0.857 0.000 EI -0.624 0.000 0.702 0.000 0.753 0.000 MI Cell Contents: Pearson correlation P-Value LE HDI Ranking 0.909 0.000