Exam 16: Simple Linear Regression and Correlation

NARRBEGIN: Movie Revenues Movie Revenues A financier whose specialty is investing in movie productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those movies 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 performers in the movie for ten recently released movies. Movie Cost of Two Highest Paid Performers ( \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 0.7 58 NARREND -{Movie Revenues Narrative} Estimate with 95% confidence the average gross revenue of a movie whose top two stars earn $5.0 million.

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 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(ad)=91.7\% Analysis of Variance Source 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 NARREND -{Oil Quality and Price Narrative} Plot the residuals against the predicted values $\hat { y }$ .

The following 10 observations of variables x and y were collected. 1 2 3 4 5 6 7 8 9 10 y 25 22 21 19 14 15 12 10 6 2 Find the least squares regression line, and the estimated value of y when x = 3.

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 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(ad)=91.7\% Analysis of Variance Source 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 NARREND -{Oil Quality and Price Narrative} Identify possible outliers.

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 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(ad)=91.7\% Analysis of Variance Source 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 NARREND -{Oil Quality and Price Narrative} Use the residuals to compute the standardized residuals.

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Dascriptive atafistics Variable Mear StDev SE Mear Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0208837 Rederatian Antalyis predictor Coef StDev T P Constant 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(ad)=91.7\% Analysis of Variance Source 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 NARREND -{Oil Quality and Price Narrative} Draw a histogram of the residuals.

In the first order linear regression model, the population parameters of the y-intercept and the slope are estimated, respectively, by:

If the coefficient of correlation is -0.80, then the percentage of the variation in y that is explained by the variation in x is:

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y.

NARRBEGIN: Cost of Books Cost of Books The editor of a major academic book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger books will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected books. These data are listed below. $\begin{array} { | c | c | c | } \hline \text { Book } & \text { Number af Pages } & \text { Selling Price (\$) } \\\hline 1 & 844 & 55 \\2 & 727 & 50 \\3 & 360 & 35 \\4 & 915 & 60 \\5 & 295 & 30 \\6 & 706 & 50 \\7 & 410 & 40 \\8 & 905 & 53 \\9 & 1058 & 65 \\10 & 865 & 54 \\11 & 677 & 42 \\12 & 912 & 58 \\\hline\end{array}$ NARREND -{Cost of Books Narrative} Determine the coefficient of determination and discuss what its value tells you.

If cov(x, y) = 1260, $s _ { x } ^ { 2 } = 1600$ , and $s _ { y } ^ { 2 } = 1225$ , then the coefficient of determination is:

The unbiased estimator of the variance of the error variable is found by taking ____________________ divided by n - 2.

A medical statistician wanted to examine the relationship between the amount of sunshine (x) and incidence of skin discolorations (y). As an experiment he found the number of skin discolorations detected per 100,000 of population and the average daily sunshine in eight counties around the country. These data are shown below. Average Daily Sunshine 5 7 6 7 8 6 4 3 Slin Disculorations per 100,000 7 11 9 12 15 10 7 5 Predict with 95% confidence the skin discolorations per 100,000 in a county with a daily average of 6.5 hours of sunshine.

NARRBEGIN: Marc Anthony Concert 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: SUMMARY OUTPUT DESCRIPTIVE STATISTICS Reqression Statistics Multiple R 0.80203 R Square 0.64326 Adjusted R Square 0.62344 Standard Error 0.93965 Observations 20 Age Concerts Mean 53 Mean 3.65 Standard Error 2.1849 Standard Error 0.3424 Standard Deviation 9.7711 Standard Deviation 1.5313 Sample Variance 95.4737 Samplevariance 2.3447 Count 20 Court 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ df SS MS F Sign\&icance F Regression 1 28.65711 28.65711 32.45653 2.1082-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standard Error t Stat Rvale Lower 95\% Upoer 95\% Intercept -3.01152 1.18802 -2.53491 0.02074 -5.50746 -0.5156 Age 0.12569 0.02206 5.69706 0.00002 0.07934 0.1720 NARREND -{Marc Anthony Concert Narrative} Does it appear that heteroscedasticity is a problem? Explain.

If the variance of the errors is constant for each predicted y value, the condition is called ____________________.

NARRBEGIN: UV's and Skin Cancer U V's and Skin Cancer A medical statistician wanted to examine the relationship between the amount of UV's (x) and incidence of skin cancer (y). As an experiment he found the number of skin cancers detected per 100,000 of population and the average daily sunshine in eight states around the country. These data are shown below. Average Daily UV's 5 7 6 7 8 6 4 3 Skin Cancer per 100,000 7 11 9 12 15 10 7 5 NARREND -{UV's and Skin Cancer Narrative} Calculate the standard error of estimate, and describe what this statistic tells you about the regression line.

NARRBEGIN: Truck Speed & Gas Mileage Truck Speed and Gas Mileage An economist wanted to analyze the relationship between the speed of a truck (x) and its gas mileage (y). As an experiment a truck is operated at several different speeds and for each speed the gas mileage is measured. These data are shown below. Speed 25 35 45 50 60 65 70 Gas Mileage 40 39 37 33 30 27 25 NARREND -{Truck Speed and Gas Mileage Narrative} Calculate the standard error of estimate, and describe what this statistic tells you about the regression line.

NARRBEGIN: Oil Quality and Price 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. $\begin{array} { | c | c | } \hline \text { Oil degrees API } & \text { Price per barrel (in \$) } \\\hline 27.0 & 12.02 \\28.5 & 12.04 \\30.8 & 12.32 \\31.3 & 12.27 \\31.9 & 12.49 \\34.5 & 12.70 \\34.0 & 12.80 \\34.7 & 13.00 \\37.0 & 13.00 \\41.0 & 13.17 \\41.0 & 13.19 \\38.8 & 13.22 \\39.3 & 13.27 \\\hline\end{array}$ A partial Minitab output follows: Descriptive Statistics Variable Mean StDev SE Mean Degrees 13 34.60 4.613 1.280 Price 13 1270 0.757 0.127 Bavarifinces Degrees Price Degrees 21.281667 Price 2.026750 0.208833 $\text { Regression Analysis }$ Predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 0.008220 11.59 0.000 $\mathrm{S}=0.1314 \quad \mathrm{R}-\mathrm{Sq}=92.46 \% \quad \mathrm{R}-\mathrm{Sg}(\mathrm{adj})=91.7 \%$ $\text { Analysis of Variance }$ Source DF SS MS F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Interpret the value of the slope of the regression line.

Consider the following data values of variables x and y. 2 4 6 8 10 13 7 11 17 21 27 36 a.Calculate the coefficient of determination, and describe what this statistic tells you about the relationship between the two variables. b.Calculate the Pearson coefficient of correlation. What sign does it have? Why? c.What does the coefficient of correlation calculated tell you about the direction and strength of the relationship between the two variables?

NARRBEGIN: Accidents & Rain Accidents and Rain A statistician investigating the relationship between the amount of rain (in inches) and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year. The results are shown below. Day Rain Number of Accidents 1 0.05 5 2 0.12 6 3 0.05 2 4 0.08 4 5 0.10 6 0.35 14 7 0.15 7 8 0.30 13 9 0.10 7 10 0.20 10 NARREND -{Accidents and Rain Narrative} Find the least squares regression line.