Exam 16: Simple Linear Regression and Correlation

The residual is defined as the difference between:

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of hardware stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Salesperson Years of Exgperience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 NARREND -{Sales and Experience Narrative} Conduct a test of the population slope to determine at the 5% significance level whether a positive linear relationship exists between years of experience and sales.

NARRBEGIN: Game Winnings & Ed.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 NARREND -{Game Winnings & Education Narrative} Plot the residuals against the predicted values $\hat { y }$ . What does the graph tell you?

NARRBEGIN: Oil Quality/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 positive 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 statistical software output follows: Dascriptive atafistics Variable Mear StDev SE Mear 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 Regression Analysis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 \square.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 Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Calculate the Pearson correlation coefficient. What sign does it have? Why?

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} For what values of API gravity degrees do we feel comfortable making predictions for oil price?

In a simple linear regression problem, the least squares line is $\tilde { y } = - 3.75 + 1.25 x$ , and the coefficient of determination is 0.81. The coefficient of correlation must be -0.90.

The regression line $\tilde { y } = 3 + 2 x$ has been fitted to the data points (4, 8), (2, 5), and (1, 2). The sum of the squared residuals will be:

The degrees of freedom for the test statistic for the slope is ____________________.

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of hardware stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Salesperson Years of Exgperience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 NARREND -{Sales and Experience Narrative} Determine the standard error of estimate and describe what this statistic tells you about the regression line.

NARRBEGIN: Sales and Experience Sales and Experience The general manager of a chain of Designer stores believes that experience is the most important factor in determining the level of success of a salesperson. To examine this belief she records last month's sales (in $1,000s) and the years of experience of 10 randomly selected salespeople. These data are listed below. Sales person Years of Experience Sales 1 0 7 2 2 9 3 10 20 4 3 15 5 8 18 6 5 14 7 12 20 8 7 17 9 20 30 10 15 25 NARREND -{Sales and Experience Narrative} Predict with 95% confidence the monthly sales of a salesperson with 10 years of experience.

In the simple linear regression model, the slope represents the:

NARRBEGIN: Oil Quality/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 positive 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 statistical software output follows: Dascriptive atafistics Variable Mear StDev SE Mear 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 Regression Analysis predictor Coef StDev T P Constant 9.4349 0.2867 32.91 0.000 Degrees 0.095235 \square.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 Regeression 1 2.3162 2.3162 134.24 0.000 Resichul Entar 11 0.1898 0.0173 Total 12 2.5060 NARREND -{Oil Quality and Price Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables.

The prediction interval for a particular value of y is always wider than the confidence interval for mean value of y, given the same data set, x value, and confidence level.

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} Predict with 90% confidence the selling price of a book with 900 pages.

NARRBEGIN: Game Show Winnings & Ed.Game Show 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 NARREND -{Game Show Winnings & Education Narrative} Determine the coefficient of determination and discuss what its value tells you about the two variables.

NARRBEGIN: Game Winnings & Ed.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 NARREND -{Game Winnings & Education Narrative} Identify possible outliers.

In simple linear regression, which of the following statements indicates there is no linear relationship between the variables x and y?

In regression analysis, the residuals represent the:

NARRBEGIN: Allman Brothers Concert Allman Brothers Concert At a recent Allman Brothers 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 Reqression Statistics Mutiple 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 Sianificance 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 tStat Pvalue 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 -{Allman Brothers Concert Narrative} Interpret the value of the slope of the regression line.

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} Conduct a test of the population coefficient of correlation to determine at the 5% significance level whether a positive linear relationship exists between payment to the two highest-paid performers and gross revenue.