Exam 16: Simple Linear Regression and Correlation

The farther a given value of x is from the mean of x, the ____________________ the estimated error becomes.

If the regression line is horizontal, then we conclude that y ____________________ (is/is not) related to x.

NARRBEGIN: Wayne Newton Concert Wayne Newton Concert At a recent Wayne Newton 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 of 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 Regression 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 Sample Variance 2.3447 Count 20 Count 20 $\text { SPEARMAN RANK CORRELATION COEFFICIENT }=0.8306$ $\text { ANOVA }$ df SS MS F Signficance F Regression 1 28.65711 28.65711 32.45653 2.1082-05 Residual 18 15.89289 0.88294 Total 19 44.55 Coefficients Standerd Emor t Stat Rvalie Lower 95\% Uoper 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 -{Wayne Newton Concert Narrative} Predict with 95% confidence the number of concerts attended by a 45 years-old individual.

We can often detect autocorrelation by graphing the residuals against ____________________.

NARRBEGIN: Speed vs Gas Mileage Speed vs Gas Mileage An economist wanted to analyze the relationship between the speed of a car (x) and its gas mileage (y). As an experiment a car 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 -{Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.

A regression analysis between weight (y in pounds) and height (x in inches) resulted in the following least squares line: $\hat { y } = 120 + 5 x$ . This implies that if the height is increased by 1 inch, the weight, on average, is expected to:

Given that $s _ { x } ^ { 2 } = 500 , s _ { y } ^ { 2 } = 750 , \operatorname { cov } ( x , y ) = 100$ and n = 6, the standard error of estimate is:

The coefficient of correlation is used to determine:

NARRBEGIN: Sunshine and Melanoma Sunshine and Melanoma A medical researcher wanted to examine the relationship between the amount of sunshine (x) in hours, and incidence of melanoma, a type of skin cancer (y). As an experiment he found the number of melanoma cases 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 Melanoma per 100,000 7 11 9 12 15 10 7 5 NARREND -{Sunshine and Melanoma Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it.

NARRBEGIN: Comedy Shows Revenues 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 Two 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 0.7 58 NARREND -{Comedy Shows Revenues Narrative} Use the predicted and actual values of y to calculate the residuals.

NARRBEGIN: Trivia Games & Ed.Trivia Games & 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 NARREND -{Trivia Games & Education Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.

In a simple linear regression model b₁ is the ____________________ of the straight line.

The coefficient of determination is equal to the coefficient of correlation squared.

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing H₀: $\beta$ ₁ = 0 has a Student t-distribution with n -2 degrees of freedom.

NARRBEGIN: Rock Concert Revenues Rock Concert Revenues A financier whose specialty is investing in rock concerts has observed that, in general, concerts with "big-name" stars seem to generate more revenue than those concerts 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 concert for ten concert tours. Concert Cost of Twa 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 -{Rock Concert Revenues Narrative} Do the $\rho$ and $\beta$ ₁ tests in the previous questions provide the same results? Explain.

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. $\begin{array} { | c | c | c | } \hline \text { Book } & \text { Number af Peges } & \text { Selling Price (\$) } \\\hline 1 & 844 & 55 \\2 & 727 & 50 \\3 & 360 & 35 \\4 & 915 & 60 \\5 & 295 & 30 \\6 & 706 & 50 \\7 & 410 & 40 \\8 & 905 & 53 \\9 & 1058 & 65 \\10 & 865 & 54 \\11 & 677 & 42 \\12 & 912 & 58 \\\hline\end{array}$ NARREND -{Cost of Books Narrative} Estimate with 90% confidence the average selling price of all books with 900 pages.

A store manager gives a pre-employment examination to new employees. The test is scored from 1 to 100. He has data on their sales at the end of one year measured in dollars. He wants to know if there is any linear relationship between pre-employment examination score and sales. An appropriate test to use is the t-test of the population correlation coefficient.

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 $\text { Covariances }$ Degrees Price Degrees 21.281667 Price 2.026750 0208837 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 S=0.1314R-Sq=92.46\%R-Sq(adj)=91.7\% Analysis of Variance Source DF SS MS F P Regression 1 2.3162 2.3162 134.24 0.000 Residual Error 11 0.1898 0.017 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Estimate with 95% confidence the average oil price per barrel for an API degree of 35.

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} Use the regression equation $\hat { y } = - 3.0115 + 0.1257$ to determine the predicted values of y.

SSE stands for ____________________ of squares for ____________________.