Exam 11: Simple Linear Regression

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 2007, 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 Calculate the test statistic for determining whether the market model is useful for predicting daily rate of return of Company A's stock.

A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the Appraiser decided to fit the linear regression model: $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ where $y =$ appraised value of the house (in thousands of dollars) and $x =$ number of rooms. Using data collected for a sample of $n = 74$ houses in East Meadow, the following result was obtained: $\hat { y } = 74.80 + 19.72 x$ Which of the following statements concerning the deterministic model, $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ is true?

(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, give a practical interpretation of $r = .81$

The least squares model provides very good estimates of y for values of x far outside the range of x values contained in the sample.

In a comprehensive road test for new car models, one variable measured is the time it takes the 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: y = Elapsed time (in seconds) from 0 mph to 60 mph MAX: $x =$ 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 REGRESSION 1 374.285 374.285 290.83 0.0000 RESIDUAL 127 163.443 1.28695 TOTAL 128 537.728 CASES INCLUDED 129 $\quad$ MISSING CASES 0 $\text { Find and interoret the estimate } \hat{\beta}_{1} \text { in the printout above. }$

To investigate the relationship between yield of potatoes, y, and level of fertilizer application, x, a researcher divides a field into eight plots of equal size and applies differing amounts of fertilizer to each. The yield of potatoes (in pounds) and the fertilizer application (in pounds) are recorded for each plot. The data are as follows: x 1 1.5 2 2.5 3 3.5 4 4.5 y 25 31 27 28 36 35 32 34 Summary statistics yield $S S _ { x x } = 10.5 , S S _ { y y } = 112 , S S _ { x y } = 25$ , and $S S E = 52.476$ . Calculate the coefficient of correlation.

Graph the line that passes through the given points. - $( 2 , - 4 )$ and $( - 1,2 )$

Operations managers often use work sampling to estimate how much time workers spend on each operation. Work sampling-which involves observing workers at random points in time-was applied to the staff of the catalog sales department of a clothing manufacturer. The department applied regression to data collected for 40 randomly selected working days. The simple linear model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } ^ { x }$ was fit to the data. The printouts for the analysis are given below: TIME: $\quad y =$ Time spent (in hours) taking telephone orders during the day ORDERS: $\quad x =$ Number of telephone orders received during the day UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF TIME PREDICTOR VARIABLES COEFFICIENT STD ERROR STUDENT'S T P CONSTANT 10.1639 1.77844 5.72 0.0000 ORDERS 0.05836 0.00586 9.96 0.0000 R-SQUARED 0.7229 RESID. MEAN SQUARE (MSE) 11.6175 ADJUSTED R-SQUARED 0.7156 STANDARD DEVIATION 3.40844 SOURCE DF SS MS REGRESSION 1 1151.55 1151.55 99.12 0.0000 RESIDUAL 38 441.464 11.6175 TOTAL 39 1593.01 CASES INCLUDED 40 $\quad$ MISSING CASES 0 Conduct a test of hypothesis to determine if time spent (in hours) taking telephone orders during the day and the number of telephone orders received during the day are positively linearly related. Use $\alpha = .01$ .

Consider the following pairs of measurements: x 5 8 3 4 9 y 6.2 3.4 7.5 8.1 3.2 a. Construct a scattergram for the data. b. Use the method of least squares to model the relationship between x and y. c. Calculate SSE, s², and s. d. What percentage of the observed y-values fall within 2s of the values of $\hat{y}$ predicted by the least squares model?

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: $\quad x _ { 1 } =$ Boiler design pressure (pounds per square inch, i.e., psi) The simple linear model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ 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 Fill in the blank. At $\alpha = .01$ , there is between man-hours and pressure.

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 $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x .$ The results of the simple linear regression are provided below. $\hat { y } = 2,700 + 20 x , s = 65,2 \text {-tailed } p \text {-value } = .064 \text { (for testing } \beta _ { 1 } \text { ) }$ Interpret the $p$ -value for testing whether $\beta _ { 1 }$ exceeds 0 .

The coefficient of correlation is a useful measure of the linear relationship between two variables.

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 was created from a set of 25 data points. Which of the following is not an assumption required for the simple linear regression analysis to be valid?

Consider the data set shown below. Find the estimate of the slope of the least squares regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

State the four basic assumptions about the general form of the probability distribution of the random error ?.

A county real estate appraiser wants to develop a statistical model to predict the appraised value of houses in a section of the county called East Meadow. One of the many variables thought to be an important predictor of appraised value is the number of rooms in the house. Consequently, the appraiser decided to fit the simple linear regression model: $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ where $y =$ appraised value of the house (in thousands of dollars) and $x =$ number of rooms. Using data collected for a sample of $n = 86$ houses in East Meadow, the following results were obtained: $\hat { y } = 86.80 + 19.72 x$ What are the properties of the least squares line, $\hat { y } = 86.80 + 19.72 x$ ?

(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 31.7 456 24. Maryland 593 28.1 499 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, which of the following is not an assumption required for the simple linear regression analysis to be valid?

The data for $n = 24$ points were subjected to a simple linear regression with the results: $\hat { \beta } _ { 1 } = 0.81 \text { and } \mathrm { s } _ {\hat{ \beta 1} }= 0.12 \text {. }$ a. Test whether the two variables, $x$ and $y$ , are positively linearly related. Use $\alpha = .05$ . b. Construct and interpret a $90 \%$ confidence interval for $\beta _ { 1 }$ .

What is the relationship between diamond price and carat size? 307 diamonds were sampled and a straight-line relationship was hypothesized between $y =$ diamond price (in dollars) and $x =$ size of the diamond (in carats). The simple linear regression for the analysis is shown below: least Squares Linear Regression of PRICE Predictor Variables Coefficient Std Error T P Constant -2298.36 158.531 -14.50 0.0000 Size 11598.9 230.111 50.41 0.0000 R-Squared 0.8925 Resid. Mean Square (MSE) 1248950 Adjusted R-Squared 0.8922 Standard Deviation 1117.56 Interpret the standard deviation of the regression model.

(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 31.9 49,080 22. Pittsburgh 605 33.0 43,900 23. Georgetown 617 31.7 499 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, give a practical interpretation of $r ^ { 2 } = .66$ .