Exam 16: B: Simple Linear Regression and Correlation

Game Winnings & Education An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows. Contestant Years of Education Winnings 1 11 750 2 15 400 3 12 600 4 16 350 5 11 800 0 16 300 7 13 650 8 14 400 -{Game Winnings & Education Narrative} Use the regression equation $\hat { y } = 1735 - 89.1667 x$ to determine the predicted values of y.

Marc Anthony Concert At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Marc Anthony Concert Narrative} Use the residuals to compute the standardized residuals.

Oil Quality and Price Quality of oil is measured in API gravity degrees--the higher the degrees API,the higher the quality.The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel. $\begin{array} { | c | c | } \hline \text { Oi degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Covariances Degeres Price Degeres 21.281667 Price 2.026750 0.208933 Rederatian Antalyis predictor Caef StDev T P Cunstant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 S=0.1314R-Sq=92.46\%R-Sq(adj)=91.7\% Analysis of Variance Surce DF SS MS F P Regeression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 -{Oil Quality and Price Narrative} Use the predicted values and the actual values of y to calculate the residuals.

Marc Anthony Concert At a recent Marc Anthony concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected: Age 62 57 40 49 67 54 43 65 54 41 Number af Concerts 6 5 4 3 5 5 2 6 3 1 Age 44 48 55 60 59 63 69 40 38 52 Number of Concerts 3 2 4 5 4 5 4 2 1 3 An Excel output follows: -{Marc Anthony Concert Narrative} Plot the residuals against the predicted values $\hat { y }$ .

Data that exhibit an autocorrelation effect violate the regression assumption of independence.

If the plot of the residuals vs.the predicted values resembles a straight line with non-zero slope,then the regression line fits well.

Comedy Shows Revenues A financier whose specialty is investing in comedy shows has observed that,in general,shows with "big-name" stars seem to generate more revenue than those shows whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid comedians in the show for ten recently staged shows. Show Cost of Twa Highest Paid Comedian ( \mil ) Gross Revenue () 1 5.3 48 2 7.2 65 3 1.3 18 4 1.8 20 5 3.5 31 6 2.6 26 7 8.0 73 8 2.4 23 9 4.5 39 10 6.7 58 -{Comedy Shows Revenues Narrative} Plot the residuals against the predicted values of y.What does the graph tell you?