Exam 11: Simple Linear Regression

(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 $\hat { \beta } _ { 1 } = 228$ .

A breeder of Thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Horse Gestation period Life Length Horse Gestation period Life Length x (days) y (years) x (days) y (years) 1 416 24 5 356 22 2 279 25.5 6 403 23.5 3 298 20 7 265 21 4 307 21.5 Summary statistics yield $S S _ { x x } = 21,752 , S S _ { x y } = 236.5 , S S _ { y y } = 22 , \bar { x } = 332$ , and $\bar { y } = 22.5$ . Calculate $S S E , s ^ { 2 }$ , and $s$ .

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

A high value of the correlation coefficient $r$ implies that a causal relationship exists between $x$ and $y$ .

A realtor collected the following data for a random sample of ten homes that recently sold in her area. House Asking Price Days on Market A \ 114,500 29 B \ 149,900 16 C \ 154,700 59 D \ 159,900 42 E \ 160,000 72 F \ 165,900 45 G \ 169,700 12 \ 169,700 12 \ 171,900 39 \ 175,000 81 \ 289,900 121 a. Construct a scattergram for the data. b. Find the least squares line for the data and plot the line on your scattergram. c. Test whether the number of days on the market, y, is positively linearly related to the asking price, $x \text {. Use } \alpha = .05$

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: $\sum y = 1,134 , \quad \sum x = 3.642 , \sum y ^ { 2 } = 93,110 , \quad \sum x ^ { 2 } = .948622 , \text { and } \sum x y = 295.54$ Find the least squares prediction equation for predicting the number of games won, $y$ , using a straight-line relationship with the team's batting average, $x$ .

Calculate SSE and $s ^ { 2 } \text { for } n = 25 , \sum \mathrm { y } ^ { 2 } = 950 , \quad \sum \mathrm { y } = 65 , \mathrm { SS } _ { \mathrm { xy } } = 3000 , \text { and } \hat { \beta } 1 = .2$

$( - 10 , - 10 ) \text { and } ( 5,5 )$

If a least squares line were determined for the data set in each scattergram, which would have the smallest variance?

Is there a relationship between the raises administrators at County University receive and their performance on the job? A faculty group wants to determine whether job rating $( x )$ is a useful linear predictor of raise (y). Consequently, the group considered the linear regression model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ The faculty group obtained the following prediction equation: $\hat { y } = 14,000 - 2,000 x$ Which of the following statements about the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ is correct?

Consider the following pairs of observations: x 2 3 5 5 6 y 1.3 1.6 2.1 2.2 2.7 a. Construct a scattergram for the data. Does the scattergram suggest that $y$ is positively linearly related to $x$ ? b. Find the slope of the least squares line for the data and test whether the data provide sufficient evidence that $y$ is positively linearly related to $x$ . Use $\alpha = .05$ .

Graph the line that passes through the given points. - $( 0,5 ) \text { and } ( 5,0 )$

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 $\quad\quad\quad \quad 0.6027 \quad$ Resid. Mean Square (MSE) $532.986$ Adjusted R-Squared $\quad0.5972\quad$ Standard Deviation $\quad\quad\quad 23.0865$ In addition, we are told that the coefficient of correlation was calculated to be $r = 0.7763$ . Interpret this result.

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: $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x ^ { \prime }$ 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.

In team-teaching, two or more teachers lead a class. A researcher tested the use of team-teaching in mathematics education. Two of the variables measured on each teacher in a sample of 171 mathematics teachers were years of teaching experience x and reported success rate y (measured as a percentage) of team-teaching mathematics classes. The correlation coefficient for the sample data was reported as $r = - 0.32$ . Interpret this result.

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 estimated slope of the regression line.

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 the age of the warthog (in days) are listed below: Number of Grunts Age (days) 102 130 80 146 51 160 56 165 75 172 52 179 74 188 29 194 34 200 a. Find $S S E , s ^ { 2 }$ , and $s$ . b. Interpret the value of $s$ .

A company keeps extensive records on its new salespeople on the premise that sales should increase with experience. A random sample of seven new salespeople produced the data on experience and sales shown in the table. Months on Job Monthly Sales y (\ thousands) 2 2.4 4 7.0 8 11.3 12 15.0 1 .8 5 3.7 9 12.0 Summary statistics yield $S S _ { x x } = 94.8571 , S S _ { x y } = 124.7571 , S S _ { y y } = 176.5171 , \bar { x } = 5.8571$ , and $\bar { y } = 7.4571$ . State the assumptions necessary for predicting the monthly sales based on the linear relationship with the months on the job.

The equation for a (deterministic) straight line is $y = \beta _ { 0 } + \beta _ { 1 } x$ . If the line passes through the points $( 5,2 )$ and $( 9,4 )$ , find the values of $\beta _ { 0 }$ and $\beta _ { 1 }$ , respectively.