Exam 11: Simple Linear Regression

A large national bank charges local companies for using their services. A bank official reported the results of a regression analysis designed to predict the bank's charges $( y )$ , measured in dollars per month, for services rendered to local companies. One independent variable used to predict service charge to a company is the company's sales revenue $( x )$ , measured in $\$$ million. Data for 21 companies who use the bank's services were used to fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ Suppose a $95 \%$ confidence interval for $\beta 1$ is $( 15,25 )$ . Interpret the interval.

A low value of the correlation coefficient r implies that $x \text { and } y$ are unrelated.

Consider the following model $y = \beta _ { 0 } + \beta _ { 1 } x + \mathrm { E } _ { \text {where } } \mathrm { y }$ is the daily rate of return of a stock, and $x$ is the daily rate of return of the stock market as a whole, measured by the daily rate of return of Standard \& Poor's (S\&P) 500 Composite Index. Using a random sample of $n = 12$ days from 1980, the least squares lines shown in the table below were obtained for four firms. The estimated standard error of $\hat { \beta } _ { 1 }$ is shown to the right of each least squares prediction equation. Firm Estimated Market Model Estimated Standard Error of \beta1 Company A y=.0010+1.40x .03 Company B y=.0005-1.21x .06 Company C y=.0010+1.62x 1.34 Company D y=.0013+.76x .15 For which of the three stocks, Companies B, C, or D, is there evidence (at $\alpha = .05$ ) of a positive linear relationship between $y$ and $x$ ?

Is the number of games won by a major league baseball team in a season related to the team's batting average? Data from 14 teams were collected and the summary statistics yield: $\sum y = 1,134 , \sum x = 3.642 , \sum y ^ { 2 } = 93,110 , \sum x ^ { 2 } = .948622 , \text { and } \sum x y = 29$ Assume $\hat { \beta } 1 = 455.27$ . Estimate and interpret the estimate of $\sigma$ . 54

Consider the following pairs of observations: x 2 3 5 5 6 y 1.3 1.6 2.1 2.2 2.7 Find and interpret the value of the coefficient of correlation.

The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA $( x )$ and starting salary ( $y$ , in \$thousands) for each graduate were used to fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ The results of the simple linear regression are provided below. =4.25+2.75x S=5.15,S=1.87 SSyy=15.17,SSE=1.0075 Calculate the value of $r ^ { 2 }$ , the coefficient of determination.

In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and and the age of the warthog (in days) are listed below: Number of Grunts Age (days) 87 122 65 138 36 152 41 157 60 164 37 171 59 180 14 186 17 192 Find and interpret the value of r.

(Situation P) Below are the results of a survey of America's best graduate and professional schools. The top 25 business schools, as determined by reputation, student selectivity, placement success, and graduation rate, are listed in the table. For each school, three variables were measured: (1) GMAT score for the typical incoming student; (2) student acceptance rate (percentage accepted of all students who applied); and (3) starting salary of the typical graduating student. School GMAT Acc. Rate Salary 1. Harvard 644 15.0\% \ 63,000 2. Stanford 665 10.2 60,000 3. Penn 644 19.4 55,000 4. Northwestern 640 22.6 54,000 5. MIT 650 21.3 57,000 6. Chicago 632 30.0 55,269 7. Duke 630 18.2 53,300 8. Dartmouth 649 13.4 52,000 9. Virginia 630 23.0 55,269 10. Michigan 620 32.4 53.300 11. Columbia 635 37.1 52,000 12. Cornell 648 14.9 50,700 13. CMU 630 31.2 52,050 14. UNC 625 15.4 50,800 15. Cal-Berkeley 634 24.7 50,000 16. UCLA 640 20.7 51,494 17. Texas 612 28.1 43,985 18. Indiana 600 29.0 44,119 19. NYU 610 35.0 53,161 20. Purdue 595 26.8 43,500 21. USC 610 31.9 49,080 22. Pittsburgh 605 33.0 43,500 23. Georgetown 617 31.7 45,156 24. Maryland 593 28.1 42,925 25. Rochester 605 35.9 44,499 The academic advisor wants to predict the typical starting salary of a graduate at a top business school using GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using the 25 data points in the table are shown below. $\beta _ { 0 } = - 92040 \quad \hat { \beta } _ { 1 } = 228 \quad s = 3213 \quad r ^ { 2 } = .66 \quad r = .81 \quad \mathrm { df } = 23 \quad t = 6.67$ -For the situation above, write the equation of the probabilistic model of interest.

A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000's) of graduates of the program based on the amount of tuition (in $1000's) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Coefficient Std Error T P Constant 18.1849 10.3336 1.76 0.0826 Size 1.47494 0.14017 10.52 0.0000 R-Squared 0.6027 Resid. Mean Square (MSE) 532.986 Adjusted R-Squared 0.5972 Standard Deviation 23.0865 The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000. The results are shown here: 95% confidence interval for E(Y): ($123,390, $134,220) 95% prediction interval for Y: ($82,476, $175,130) Which of the following interpretations is correct if you want to use the model to predict Y for a single MBA programs?

(Situation P) Below are the results of a survey of America's best graduate and professional schools. The top 25 business schools, as determined by reputation, student selectivity, placement success, and graduation rate, are listed in the table. for each school, three variables were measured: (1) GMAT score for the typical incoming student; (2) student acceptance rate (percentage accepted of all students who applied); and (3) starting salary of the typical graduating student. School GMAT Acc. Rate Salary 1. Harvard 644 15.0\% \ 63,000 2. Stanford 665 10.2 60,000 3. Penn 644 19.4 55,000 4. Northwestern 640 22.6 54,000 5. MIT 650 21.3 57,000 6. Chicago 632 30.0 55,269 7. Duke 630 18.2 53,300 8. Dartmouth 649 13.4 52,000 9. Virginia 630 23.0 55,269 10. Michigan 620 32.4 53.300 11. Columbia 635 37.1 52,000 12. Cornell 648 14.9 50,700 13. CMU 630 31.2 52,050 14. UNC 625 15.4 50,800 15. Cal-Berkeley 634 24.7 50,000 16. UCLA 640 20.7 51,494 17. Texas 612 28.1 43,985 18. Indiana 600 29.0 44,119 19. NYU 610 35.0 53,161 20. Purdue 595 26.8 43,500 21. USC 610 31.9 49,080 22. Pittsburgh 605 33.0 43,500 23. Georgetown 617 31.7 45,156 24. Maryland 593 28.1 42,925 25. Rochester 605 35.9 44,499 The academic advisor wants to predict the typical starting salary of a graduate at a top business school using GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using the 25 data points in the table are shown below. $\beta _ { 0 } = - 92040 \quad \hat { \beta } _ { 1 } = 228 \quad s = 3213 \quad r ^ { 2 } = .66 \quad r = .81 \quad \mathrm { df } = 23 \quad t = 6.67$ -For the situation above, write the equation of the least squares line.

Consider the data set shown below. Find the coefficient of correlation for between the variables x and y. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

The dean of the Business School at a small Florida college wishes to determine whether the grade-point average (GPA) of a graduating student can be used to predict the graduate's starting salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting salaries. Records for 23 of last year's Business School graduates are selected at random, and data on GPA $( x )$ and starting salary $( y$ , in \$thousands) for each graduate were used to fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ The results of the simple linear regression are provided below. =4.25+2.75x, SSxy=5.15,SSxx=1.87 SSyy =15.17,SSE=1.0075 Range of the x -values: 2.23-3.85 Range of the y-values: 9.3-15.6 Suppose a $95 \%$ prediction interval for $y$ when $x = 3.00$ is $( 16,21 )$ . Interpret the interval.

The probabilistic model allows the E(y) values to fall around the regression line while the actual values of y must fall on the line.

An academic advisor wants to predict the typical starting salary of a graduate at a top business school using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 s = 3213 r ^ { 2 } = .66 r = .81 \mathrm { df } = 23 t = 6.67$ Give a practical interpretation of $r ^ { 2 } = .66$ .

In team-teaching, two or more teachers lead a class. An researcher tested the use of team-teaching in mathematics education. Two of the variables measured on each sample of 182 mathematics teachers were years of teaching experience x and reported success rate y (measured as a percentage) of team-teaching mathematics classes. a. The researcher hypothesized that mathematics teachers with more years of experience will report higher perceived success rates in team-taught classes. State this hypothesis in terms of the parameter of a linear model relating x to y. b. The correlation coefficient for the sample data was reported as $\mathrm { r } = - 0.28$ . Interpret this result. c. Does the value of r support the hypothesis? Test using $\alpha = .05$

An academic advisor wants to predict the typical starting salary of a graduate at a top business school using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 \mathrm {~s} = 3213 r ^ { 2 } = .66 r = .81 \mathrm { df } = 23 \quad t = 6.67$ Give a practical interpretation of $r = .81$ .

In a comprehensive road test on new car models, one variable measured is the time it takes a car to accelerate from 0 to 60 miles per hour. To model acceleration time, a regression analysis is Conducted on a random sample of 129 new cars. TIME60: $\quad y =$ Elapsed time (in seconds) from $0 \mathrm { mph }$ to $60 \mathrm { mph }$ MAX: $\quad x =$ Maximum speed attained (miles per hour) The simple linear model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to the data. Computer printouts for the analysis are given below: NWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME60 PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 18.7171 0.63708 29.38 0.0000 MAX -0.08365 0.00491 -17.05 0.0000 R-SQUARED 0.6960 RESID. MEAN SQUARE (MSE) 1.28695 ADJUSTED R-SQUARED 0.6937 STANDARD DEVIATION 1.13444 SOURCE DF SS MS F P REGRESSION 1 374.285 374.285 290.83 0.0000 RESIDUAL 127 163.443 1.28695 TOTAL 128 537.728 CASES INCLUDED 129 MISSING CASES 0 Fill in the blank: "At $\alpha = .05$ , there is between maximum speed and acceleration time."

A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following two variables: MANHRS: $\quad y =$ Number of man-hours required to erect the drum PRESSURE: $\quad x =$ Boiler design pressure (pounds per square inch, i.e., $\mathrm { psi }$ ) The simple linear model $E ( y ) = \beta _ { 1 } + \beta _ { 1 } x$ was fit to the data. A printout for the analysis appears below: UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 1.88059 0.58380 3.22 0.0028 PRESSURE 0.00321 0.00163 2.17 0.0300 R-SQUARED 0.4342 RESID. MEAN SQUARE (MSE) 4.25460 ADJUSTED R-SQUARED 0.4176 STANDARD DEVIATION 2.06267 SOURCE DF SS MS REGRESSION 1 111.008 111.008 5.19 0.0300 RESIDUAL 34 144.656 4.25160 TOTAL 35 255.665 Give a practical interpretation of the coefficient of determination, $r ^ { 2 }$ .

A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000's) of graduates of the program based on the amount of tuition (in $1000's) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Coefficient Std Error Constant 18.1849 10.3336 1.76 0.0826 Size 1.47494 0.14017 10.52 0.0000 R-Squared 0.6027 Resid. Mean Square (MSE) 532.986 Adjusted R-Squared 0.5972 Standard Deviation 23.0865 The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the tuition charged by the MBA program was $75,000. The results are shown here: 95% confidence interval for E(Y): ($123,390, $134,220) 95% prediction interval for Y: ($82,476, $175,130) Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all MBA programs?