Exam 11: Simple Linear Regression

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

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) 90 125 68 141 39 155 44 160 63 167 40 174 62 183 17 189 20 195 Find and interpret the value of r.

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 Fill in the blank. At $\alpha = 0.05$ , there is_____ between the amount of tuition charged by an MBA program and the average starting salary of graduates of the program.

Consider the data set shown below. Find the coefficient of determination for the simple linear regression model. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

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 Interpret the estimated slope of the regression line.

Consider the following model $y = \beta _ { 0 } + \beta _ { 1 } x + \epsilon$ , where $y$ 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 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.

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

A large national bank charges local companies for using its services. A bank official reported the results of a regression analysis designed to predict the bankʹs charges (y), measured in dollars per Month, for services rendered to local companies. One independent variable used to predict the Service charge to a company is the companyʹs sales revenue (x), measured in $ million. Data for 21 Companies who use the bankʹs services were used to fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x .$ The results of the simple linear regression are provided below. $\hat { y } = 2,700 + 20 x$ Interpret the estimate of $\beta _ { 0 }$ , the $y$ -intercept of the line.

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 graduateʹs starting Salary. More specifically, the dean wants to know whether higher GPAs lead to higher starting Salaries. Records for 23 of last yearʹs 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 value of the test statistic for testing $\beta 1$ is 17.169. Select the proper conclusion.

The equation for a (deterministic) straight line is $y = \beta _ { 0 } + \beta _ { 1 } x$ . If the line passes through the points $( 7,6 )$ and $( 5,4 )$ , find the values of $\beta _ { 0 }$ and $\beta _ { 1 }$ , respectively.

A $95 \%$ prediction interval for SALARY when GMAT $= 600$ is $( \$ 37,915 , \$ 51,948 )$ . Interpret this interval for the situation above.

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 $0 \mathrm { mph }$ to $60 \mathrm { mph }$ 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: 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 CASES INCLUDED 129 MISSING CASES 0 $\text { Find and interpret the estimate } \hat { \beta } 1 \text { in the printout above. }$

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 Life Horse Gestation Life period Length period 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 $\bar { y } = 22.5$ . Test to determine if a linear relationship exists between the gestation period and the length of life of a horse. Use $\alpha = .05$ and use $s = 1.97$ as an estimate of $\sigma$ .

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 Life Horse Gestation Life period Length period 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 $\bar { y } = 22.5$ . Find a $95 \%$ prediction interval for the length of life of a horse that had a gestation period of 300 days. Use $s = 2$ as an estimate of $\sigma$ and use $\hat { y } = 18.89 + .01087 x$ .

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. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 \quad s = 3213 \quad \mathrm { df } = 23 \quad t = 6.67$ Set up the null and alternative hypotheses for testing whether a linear relationship exists between SALARY and GMAT.

The data for n = 25 points were subjected to a simple linear regression with the results: $\hat { \beta } _ { 1 } = 0.94 \text { and } \mathrm { s } _ { \beta 1 } ^ { \wedge } = 0.13 \text {. }$ 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 $\beta _ { 1 }$ .

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.

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 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.

Consider the following model $y = \beta _ { 0 } + \beta _ { 1 } x + \epsilon$ , where $y$ 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 lineal relationship between $y$ and $x$ ?