Exam 11: Simple Linear Regression

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.

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 $S S _ { x x } = 10.5 , S S _ { y y } = 112 , S S _ { x y } = 25 , \bar { x } = 2.75$ , and $\bar { y } = 31$ . Find the least squares prediction equation.

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.

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. We are told that the coefficient of correlation was calculated to be r = 0.7763. Use this information to calculate the test statistic that would be used to determine if a positive linear relationship exists between the two variables.

A low value of the correlation coefficient r implies that x and y are unrelated.

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?

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) 83 118 61 134 32 148 37 153 56 160 33 167 55 176 10 182 13 188 a. Write the equation of a straight-line model relating number of grunts $( y )$ to age $( x )$ . b. Give the least squares prediction equation. c. Give a practical interpretation of the value of $\hat { \beta } _ { 0 }$ , if possible. d. Give a practical interpretation of the value of $\hat { \beta } _ { 1 }$ , if possible.

(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 s = 3213.

Suppose you fit a least squares line to 25 data points and the calculated value of SSE is 0.42. a. Find $s ^ { 2 }$ , the estimator of $\sigma ^ { 2 }$ . b. What is the largest deviation you might expect between any one of the 25 points and the least squares line?

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

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. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 s = 3213 r ^ { 2 } = .66 r = .81 \mathrm { df } = 23 \quad t = 6.67$ A 95% prediction interval for SALARY when GMAT = 600 is approximately ($37,915, $51,984). Interpret this interval.

Plot the line y = 4 - 2x. Then give the slope and y-intercept of the line.

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. MEANSQUARE (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 ? =.05, there is ________________ between maximum speed and acceleration time."

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

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. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 \mathrm {~s} = 3213 \mathrm { r } ^ { 2 } = .66 \quad r = .81 \mathrm { df } = 23 \quad t = 6.67$ Give a practical interpretation of $r ^ { 2 } = .66$ . A) $66 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight -line model. B) We expect to predict SALARY to within $2 [ \sqrt { .66 } ]$ of its true value using GMAT in a straight-line model. C) We estimate SALARY to increase $\$ .66$ for every 1 -point increase in GMAT. D) We can predict SALARY correctly $66 \%$ of the time using GMAT in a straight-line model.

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 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 determination.

Locate the values of $S S E , s ^ { 2 }$ , and $s$ on the printout below. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ Model Summary Model R Square Adjusted Square Std. Error of the Estimate 1 .859 .737 .689 11.826 $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$ ANOVA Model Sum of Squares df Mean Square F Sig. 1 Regression 4512.024 1 4512.024 32.265 .001 Residual 1678.115 12 139.843 Total 6190.139 13 A) $S S E = 1678.115 ; s ^ { 2 } = 139.843 ; s = 11.826$ B) $S S E = 4512.024 ; s ^ { 2 } = 4512.024 ; s = 32.265$ C) $S S E = 6190.139 ; s ^ { 2 } = 4512.024 ; s = 32.265$ D) $S S E = 4512.024 ; s ^ { 2 } = 139.843 ; s = 11.826$

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 $n = 74$ houses in East Meadow, the following results were obtained: $\hat { y } = 74.80 + 19.72 x$ Give a practical interpretation of the estimate of the y-intercept of the least squares line.