A survey of a random sample of 48 students was conducted by a group of statistics students to investigate the relationshipbetween Sleep=average sleep time per night (in hours) and the following variables:Study =average study time per day (in hours),CoffeeCups =average number of cups of coffee consumed by a student per day,and Extra = average number of extra-curricular activities per dayFemale (1 =female, 0 = male)Age=age in yearsIn the first part of their analysis the students run the simple linear regression of Sleep on FemaleProblem 2a) Is the estimated multiple linear regression model of Sleep with allvariables listed in Problem 1 statistically significant at = 001? (State therelevant 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 thesimple regressions but are not statistically significant at = 005 in themultiple regression. Give a reason for their lack of statistical significance inthe multiple regression.f ) Before fitting the multiple regression models the students hypothesized that for every one cups increase in Co?eeCups, the expected decreasein 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 usinga hypothesis test with = 005g) They also thought that the average sleep time in the population ofNYU undergraduate student is more that 7 hours. Is there evidence in thedata, at = 005 that they were right? (State the hypothesis test, rejectionrule and your conclusion) 2 Minitab output for problems 1 and 2The regression equation isSleep = 6.62 – 0.708 Female PredictorConstantFemale Coef6.6250-0.7083 S = 1.07802 SE Coef0.22010.3112 R-Sq = 10.1% T30.11 P0.000 R-Sq(adj) = 8.2% Descriptive Statistics: SleepVariableSleep N48 N*0 Mean6.271 SE Mean0.162 StDev1.125 Minimum4.000 Q15.250 Median6.000 Q37.000 The regression equation isSleep = 5.94 + 0.0173 Age + 0.227 Study – 0.122 Extra – 0.583 Female- 0.301 Cupscoffee PredictorConstantAgeStudyExtraFemaleCupscoffee Coef5.93990.017340.22737-0.1215-0.5826-0.3009 SE Coef0.93840.044360.078820.12310.27570.1060 T6.330.392.88-0.99-2.11-2.84 P0.0000.6980.0060.3290.0410.007 MS4.78740.8462 F5.66 S = 0.919914 Analysis of VarianceSourceRegressionResidual ErrorTotal DF54247 SS 59.4792 P The regression equation isSleep = 5.98 + 0.227 Study – 0.377 Cupscoffee PredictorConstantStudyCupscoffee S = 0.957993 Coef5.97550.22720-0.3765 SE Coef0.34330.074930.1015 R-Sq = 30.6% T17.413.03-3.71 P0.0000.0040.001 R-Sq(adj) = 27.5% Maximum9.000 Correlations: Sleep, Age, Study, Extra, Cupscoffee, FemaleSleep0.2560.079 Age Study 0.3060.035 0.3730.009 Extra -0.2900.045 -0.1670.256 0.0900.542 Cupscoffee -0.4050.004 0.0040.977 0.1630.268 0.3870.007 -0.1550.293 0.0890.547 0.1560.288 Age Female Cell Contents: Pearson correlationP-Value Study Extra Cupscoffee 0.1660.259
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