Deck 11: Simple Linear Regression

Question 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

$\begin{array}{lcccll}\text {Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & {\text { P }} \\\text { Constant } & -2298.36 & 158.531 & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\\text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56\end{array}$

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the carat size of the diamond was 1 carat. The results are shown here:

95% confidence interval for E(Y): ($9091.60, $9509.40)
95% prediction interval for Y: ($7091.50, $11,510.00)

Which of the following interpretations is correct if you want to use the model to determine the price of a single 1-carat diamond?

A) We are 95% confident that the average price of all 1-carat diamonds will fall between $9091.60 and $9509.40.
B) We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00.
C) We are 95% confident that the price of a 1-carat diamond will fall between $9091.60 and $9509.40.
D) We are 95% confident that the average price of all 1-carat diamonds will fall between $7091.50 and $11,510.00.

Answer 1

We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00.

Question 2

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

$\begin{array} { l c c c r l } \text {Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\ \text { Constant } & - 2298.36 & 158.531 & - 14.50 & 0.0000 \\ \text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000 \end{array}$

$\begin{array} { l c c c } \text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\ \text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56 \end{array}$

Interpret the standard deviation of the regression model.

A) We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values.
B) We can explain 89.25% of the variation in the sampled diamond prices around their mean using the size of the diamond in a linear model.
C) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $1117.56.
D) We expect most of the sampled diamond prices to fall within $1117.56 of their least squares predicted values.

Answer 2

We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values.

Question 3

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

A)

B)
https://d2lvgg3v3hfg70.cloudfront.net/TB2969/.
C)
11ecdceb_29d2_5c1a_aa93_63192cff57a2_TB2969_11
D)

Answer 3

Question 4

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

$\begin{array} { l c c c l l }\text { Predictor}\\ \text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\ \text { Constant } & - 2298.36 & 158.531 & - 14.50 & 0.0000 \\ \text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000 \end{array}$

$\begin{array} { l c c c } \text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\ \text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56 \end{array}$

Which of the following conclusions is correct when testing to determine if the size of the diamond is a useful positive linear predictor of the price of a diamond?

A) There is sufficient evidence to indicate that the size of the diamond is a useful positive linear predictor of the price of a diamond when testing at $\alpha = 0.05$ .
B) The sample size is too small to make any conclusions regarding the regression line.
C) There is insufficient evidence to indicate that the price of the diamond is a useful positive linear predictor of the size of a diamond when testing at $\alpha = 0.05$ .
D) There is insufficient evidence to indicate that the size of the diamond is a useful positive linear predictor of the price of a diamond when testing at $\alpha = 0.05$ .

Question 5

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

$\begin{array}{lrccl}\text { Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \mathbf{P} \\\text { Constant } & 18.1849 & 10.3336 & 1.76 & 0.0826 \\\text { Size } & 1.47494 & 0.14017 & 10.52 & 0.0000\end{array}$

R-Squared $\quad\quad\quad\quad 0.6027 \quad$ Resid. Mean Square (MSE) $532.986$
Adjusted R-Squared $\quad 0.5972 \quad$ Standard Deviation $\quad\quad\quad23.0865$

Fill in the blank. At ? = 0.05, there is _________________ between the amount of tuition charged by an MBA program and the average starting salary of graduates of the program.

A) …insufficient evidence of a positive linear relationship…
B) …sufficient evidence of a positive linear relationship…
C) …sufficient evidence of a negative linear relationship…

Question 6

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
$\begin{array}{l}\text { Predictor }\\\begin{array}{lcccll}\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & {\text { P }} \\\text { Constant } & -2298.36 & 158.531 & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\\text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56\end{array}$

Interpret the coefficient of determination for the regression model.

A) There is sufficient evidence to indicate that the size of the diamond is a useful predictor of the price of a diamond when testing at alpha $= 0.05$ .
B) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $\$ 1117.56$ .
C) We can explain $89.25 \%$ of the variation in the sampled diamond prices around their mean using the size of the diamond in a linear model.
D) We expect most of the sampled diamond prices to fall within $\$ 2235.12$ of their least squares predicted values.

Question 7

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

$\begin{array}{l}\text { Predictor }\\\begin{array}{lcccll}\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & {\text { P }} \\\text { Constant } & -2298.36 & 158.531 & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\\text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56\end{array}$


Which of the following assumptions is not stated correctly?

A) The values of $\varepsilon$ associated with any two observations are dependent on one another.
B) The mean of the probability distribution of $\varepsilon$ is 0 .
C) The probability distribution of $\varepsilon$ is normal.
D) The variance of the probability distribution of $\varepsilon$ is constant for all settings of the independent variable.

Question 8

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?

A) The errors of predicting SALARY are normally distributed.
B) The errors of predicting SALARY have a mean of 0.
C) The errors of predicting SALARY have a variance that is constant for any given value of GMAT.
D) SALARY is independent of GMAT.

Question 9

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

$\begin{array}{lccccc}\text { Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\\text { Constant } & -2298.36 & 158.531 & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\\text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56\end{array}$

The model was then used to create 95% confidence and prediction intervals for y and for E(Y) when the carat size of the diamond was 1 carat. The results are shown here:
95% confidence interval for E(Y): ($9091.60, $9509.40)
95% prediction interval for Y: ($7091.50, $11,510.00)
Which of the following interpretations is correct if you want to use the model to estimate E(Y) for all 1-carat diamonds?

A) We are 95% confident that the average price of all 1-carat diamonds will fall between $7091.50 and $11,510.00.
B) We are 95% confident that the price of a 1-carat diamond will fall between $7091.50 and $11,510.00.
C) We are 95% confident that the price of a 1-carat diamond will fall between $9091.60 and $9509.40.
D) We are 95% confident that the average price of all 1-carat diamonds will fall between $9091.60 and $9509.40.

Question 10

Question 11

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

$\begin{array}{lrrcl}\text {Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\\text { Constant } & 18.1849 & 10.3336 & 1.76 & 0.0826 \\\text { Size } & 1.47494 & 0.14017 & 10.52 & 0.0000\end{array}$

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) There is almost no linear relationship between the amount of tuition charged and the average starting salary variables.
B) There is a very weak positive linear relationship between the amount of tuition charged and the average starting salary variables.
C) There is a fairly strong positive linear relationship between the amount of tuition charged and the average starting salary variables.
D) There is a fairly strong negative linear relationship between the amount of tuition charged and the average starting salary variables.

Question 12

Locate the values of $S S E , s ^ { 2 }$ , and $s$ on the printout below.

$\begin{array}{l}\text {\quad\quad\quad\quad\quad\quad\quad Model Summary }\\\begin{array}{|c|c|c|c|c|}\hline \text { Model } & \mathrm{R} & \mathrm{R} \text { Square } & \begin{array}{c}\text { Adjusted } \\\text { R Square }\end{array} & \begin{array}{c}\text { Std. Error of } \\\text { the Estimate }\end{array} \\\hline 1 & .859 & .737 & .689 & 11.826 \\\hline\end{array}\end{array}$

$\begin{array}{l}\text {\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad ANOVA }\\\begin{array}{|c|l|c|c|c|c|c|}\hline \text { Model } & & \text { Sum of Squares } & \text { df } & \text { Mean Square } & \text { F } & \text { Sig. } \\\hline 1 & \text { Regression } & 4512.024 & 1 & 4512.024 & 32.265 & .001 \\\hline & \text { Residual } & 1678.115 & 12 & 139.843 & & \\\hline & \text { Total } & 6190.139 & 13 & & & \\\hline\end{array}\end{array}$

A) $S S E = 4512.024 ; s ^ { 2 } = 4512.024 ; s = 32.265$
B) $S S E = 6190.139 ; s ^ { 2 } = 4512.024 ; s = 32.265$
C) $S S E = 1678.115 ; s ^ { 2 } = 139.843 ; s = 11.826$
D) $S S E = 4512.024 ; s ^ { 2 } = 139.843 ; s = 11.826$

Question 13

Question 14

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

$\begin{array} { l r c c l }\text { Predictor}\\ \text { Variables } & \text { Coefficient } & \text { Std Error } & \mathbf { T } & \mathbf { P } \\ \text { Constant } & 18.1849 & 10.3336 & 1.76 & 0.0826 \\ \text { Size } & 1.47494 & 0.14017 & 10.52 & 0.0000 \\ & & & \\ \text { R-Squared } & 0.6027 & \text { Resid. Mean Square (MSE) } & 532.986 \\ \text { Adjusted R-Squared } &0.5972 & \text { Standard Deviation } & 23.0865 \end{array}$

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?

A) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $82,476 and $175,130.
B) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $123,390 and $134,220.
C) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $82,476 and $175,130.
D) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $123,390 and $134,220.

Question 15

Consider the data set shown below. Find the estimate of the y-intercept of the least squares regression line. $$\begin{array} { c | c | c | c | c | c | c | c } \mathrm { y } & 0 & 3 & 2 & 3 & 8 & 10 & 11 \\\hline \mathrm { x } & - 2 & 0 & 2 & 4 & 6 & 8 & 10\end{array}$$

A) 1.49045
B) 0.9003
C) 0.94643
D) 1.5

Question 16

What is the relationship between diamond price and carat size? 307 diamonds were sampled (ranging in size from $0.18$ to $1.1$ carats) 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

$\begin{array}{l}\text { Predictor }\\\begin{array}{lccccl}\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\\text { Constant } & -2298.36 & 158.531 & & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}\end{array}$

$\begin{array}{llll}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950\\\text { Adiusted R-Souared } & 08920 & \text { Standard Deviation } & 111756\end{array}$
Interpret the estimated y-intercept of the regression line.

A) No practical interpretation of the y-intercept exists since a diamond of 0 carats cannot exist and falls outside the range of the carat sizes sampled.
B) When a diamond is 11598.9 carats in size, we estimate the price of the diamond to be $2298.36.
C) When a diamond is 0 carats in size, we estimate the price of the diamond to be $2298.36.
D) When a diamond is 0 carats in size, we estimate the price of the diamond to be $11,598.90.

Question 17

Consider the data set shown below. Find the coefficient of determination for the simple linear regression model.

A) 0.9003
B) 0.9489
C) 0.8804
D) 0.9383

Question 18

Consider the data set shown below. Find the coefficient of correlation for between the variables x and y.

A) 0.9383
B) 0.8804
C) 0.9489
D) 0.9003

Question 19

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

$\begin{array}{lrccl}\text {Predictor}\\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\\text { Constant } & 18.1849 & 10.3336 & 1.76 & 0.0826 \\\text { Size } & 1.47494 & 0.14017 & 10.52 & 0.0000\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.6027 & \text { Resid. Mean Square (MSE) } & 532.986 \\\text { Adjusted R-Squared } & 0.5972 & \text { Standard Deviation } & 23.0865\end{array}$

Interpret the estimated slope of the regression line.

A) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will decrease by $1474.94.
B) For every $1000 increase in the average starting salary, we estimate that the tuition charged by the MBA program will increase by $1474.94.
C) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $1474.94.
D) For every $1474.94 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $18,184.90.

Question 20

Consider the data set shown below. Find the estimate of the slope of the least squares regression line.

A) 1.49045
B) 1.5
C) 0.9003
D) 0.94643

Question 21

Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line.

$\begin{array}{c|c|c|c|c|c|c|c}\mathrm{y} & 0 & 3 & 2 & 3 & 8 & 10 & 11 \\\hline \mathrm{x} & -2 & 0 & 2 & 4 & 6 & 8 & 10\end{array}$

A) $0.94643 \pm 0.27603$
B) $0.94643 \pm 0.33306$
C) $0.94643 \pm 0.36203$
D) $0.94643 \pm 0.28377$

Question 22

Suppose you fit a least squares line to 23 data points and the calculated value of SSE is $0.495$ .
a. Find $s ^ { 2 }$ , the estimator of $\sigma ^ { 2 }$ .
b. What is the largest deviation you might expect between any one of the 23 points and the least squares line?

Question 23

Consider the data set shown below. Find the standard deviation of the least squares regression line.

A) 1.5
B) 1.49045
C) 0.9003
D) 0.94643

Question 24

Consider the following pairs of measurements:

Question 25

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, $x$ .

Question 26

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:

$\begin{array}{l|cccccccc}\hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 \\\hline y & 25 & 31 & 27 & 28 & 36 & 35 & 32 & 34 \\\hline\end{array}$
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.

Question 27

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:
$$\begin{array}{l}\text { Least Squares Linear Regression of Salary }\\\begin{array} { l r c c l } \text { Predictor } & & & & \\\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\\text { Constant } & 18.1849 & 10.3336 & 1.76 & 0.0826 \\\text { Size } & 1.47494 & 0.14017 & 10.52 & 0.0000 \\& & & \\\text { R-Squared } & 0.6027 & \text { Resid. Mean Square (MSE) } & 532.986 \\\text { Adjusted R-Squared } &0.5972 & \text { Standard Deviation } & 23.0865\end{array}\end{array}$$
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?

A) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $82,476 and $175,130.
B) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $82,476 and $175,130.
C) We are 95% confident that the average starting salary for graduates of a single MBA program that charges $75,000 in tuition will fall between $123,390 and $134,220.
D) We are 95% confident that the average of all starting salaries for graduates of all MBA programs that charge $75,000 in tuition will fall between $123,390 and $134,220.

Question 28

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.

$\begin{array}{c|c}\text { Months on Job } & \begin{array}{c}\text { Monthly Sales } \\\text { y (\$ thousands) }\end{array} \\\hline 2 & 2.4 \\4 & 7.0 \\8 & 11.3 \\12 & 15.0 \\1 & .8 \\5 & 3.7 \\9 & 12.0\end{array}$

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 $\bar { y } = 7.4571$ . Calculate a $90 \%$ confidence interval for $E ( y )$ when $x = 5$ months. Assume $s = 1.577$ and the prediction equation is $\hat { y } = - .25 + 1.315 x$ .

Question 29

Consider the following pairs of observations:
$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 0 & 3 & 3 & 5 \\\hline y & 1 & 3 & 4 & 6 & 7 \\\hline\end{array}$

a. Construct a scattergram for the data.
b. Find the least squares line, and plot it on your scattergram.
c. Find a $99 \%$ confidence interval for the mean value of $y$ when $x = 1$ .
d. Find a $99 \%$ prediction interval for a new value of $y$ when $x = 1$ .

Question 30

$$\text { Calculate SSE and } s ^ { 2 } \text { for } n = 30 , \mathrm { SS } _ { \mathrm { yy } } = 100 , \mathrm { SS } _ { \mathrm { xy } } = 60 \text {, and } \hat { \beta } 1 = .8 \text {. }$$

Question 31

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.

$\begin{array}{cccccc}\text { Horse } & \begin{array}{c}\text { Gestation } \\\text { period }\end{array} & \begin{array}{c}\text { Life } \\\text { Length }\end{array} & \text { Horse } & \begin{array}{c}\text { Gestation } \\\text { period }\end{array} & \begin{array}{c}\text { Life } \\\text { Length }\end{array} \\& x \text { (days) } & y \text { (years) } & & x \text { (days) } & y \text { (years) } \\1 & 416 & 24 & 5 & 356 & 22 \\2 & 279 & 25.5 & 6 & 403 & 23.5 \\3 & 298 & 20 & 7 & 265 & 21 \\4 & 307 & 21.5 & & &\end{array}$

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 $s$ .

Question 32

(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. $$\begin{array} { l l l l r } & \text { School } & \text { GMAT } & \text { Acc. Rate } & \text { Salary } \\\text { 1. } & \text { Harvard } & 644 & 15.0 \% & \$ 63,000 \\\text { 2. } & \text { Stanford } & 665 & 10.2 & 60,000 \\\text { 3. } & \text { Penn } & 644 & 19.4 & 55,000 \\\text { 4. } & \text { Northwestern } & 640 & 22.6 & 54,000 \\\text { 5. } & \text { MIT } & 650 & 21.3 & 57,000 \\\text { 6. } & \text { Chicago } & 632 & 30.0 & 55,269 \\\text { 7. } & \text { Duke } & 630 & 18.2 & 53,300 \\\text { 8. } & \text { Dartmouth } & 649 & 13.4 & 52,000 \\\text { 9. } & \text { Virginia } & 630 & 23.0 & 55,269 \\\text { 10. } & \text { Michigan } & 620 & 32.4 & 53.300 \\\text { 11. } & \text { Columbia } & 635 & 37.1 & 52,000 \\\text { 12. } & \text { Cornell } & 648 & 14.9 & 50,700 \\\text { 13. } & \text { CMU } & 630 & 31.2 & 52,050 \\\text { 14. } & \text { UNC } & 625 & 15.4 & 50,800 \\\text { 15. } & \text { Cal-Berkeley } & 634 & 24.7 & 50,000 \\\text { 16. } & \text { UCLA } & 640 & 20.7 & 51,494 \\\text { 17. } & \text { Texas } & 612 & 28.1 & 43,985 \\\text { 18. } & \text { Indiana } & 600 & 29.0 & 44,119 \\\text { 19. } & \text { NYU } & 610 & 35.0 & 53,161 \\\text { 20. } & \text { Purdue } & 595 & 26.8 & 43,500 \\\text { 21. } & \text { USC } & 610 & 31.9 & 49,080 \\\text { 22. } & \text { Pittsburgh } & 605 & 33.0 & 43,500 \\\text { 23. } & \text { Georgetown } & 617 & 31.7 & 456 \\\text { 24. } & \text { Maryland } & 593 & 28.1 & 499\end{array}$$
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?

A) The errors of predicting SALARY have a mean of 0.
B) SALARY is independent of GMAT.
C) The errors of predicting SALARY have a variance that is constant for any given value of GMAT.
D) The errors of predicting SALARY are normally distributed.

Question 33

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: $\begin{array}{l|cccccccc}\hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 \\\hline y & 25 & 31 & 27 & 28 & 36 & 35 & 32 & 34 \\\hline\end{array}$

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.

Question 34

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:

$\begin{array}{cc}\hline \text { Number of Grunts } & \text { Age (days) } \\\hline 90 & 125 \\68 & 141 \\39 & 155 \\44 & 160 \\63 & 167 \\40 & 174 \\62 & 183 \\17 & 189 \\20 & 195 \\\hline\end{array}$
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.

Question 35

Question 36

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:
$\begin{array}{l|cccccccc}\hline x & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 & 4.5 \\\hline y & 25 & 31 & 27 & 28 & 36 & 35 & 32 & 34 \\\hline\end{array}$

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.

Question 37

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?

A) A plot of the predicted appraised values $y$ against the number of rooms $x$ for the sample of houses in East Meadow would not result in a straight line.
B) In theory, a plot of the mean appraised value $E ( y )$ against the number of rooms $x$ for the entire population of houses in east Meadow would result in a straight line.
C) In theory, if the appraised values $y$ and number of rooms $x$ for the entire population of houses in East Meadow were obtained and the $( x , y )$ data points plotted, the points would fall in a straight line.
D) All of the above statements are true.

Question 38

Consider the following pairs of observations: Find and interpret the value of the coefficient of determination.

Question 39

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

$\begin{array}{l}\text { Predictor }\\\begin{array}{lcccll}\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & {\text { P }} \\\text { Constant } & -2298.36 & 158.531 & & -14.50 & 0.0000 \\\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000\end{array}\end{array}$

$\begin{array}{lccc}\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \\\text { Adjusted R-Squared } & 0.8922 & \text { Standard Deviation } & 1117.56\end{array}$


Interpret the estimated slope of the regression line.

A) For every $\$ 1$ decrease in the price of the diamond, we estimate that the size of the diamond will increase by $11,598.9$ carats.
B) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $\$ 11,598.90$ .
C) For every 2298.36-carat decrease in the size of a diamond, we estimate that the price of the diamond will increase by $\$ 11,598.90$ .
D) For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will decrease by $\$ 2298.36$ .

Question 40

Construct a 95% confidence interval for $\beta _ { 1 } \text { when } \hat { \beta } 1 = 49 , s = 4 , \mathrm { SS } _ { \mathrm { Xx } } = 55 \text {, and } n = 15 \text {. }$

Question 41

A realtor collected the following data for a random sample of ten homes that recently sold in her area.

$$\begin{array} { | c | c | c | } \hline \text { House } & \text { Asking Price } & \text { Days on Market } \\\hline \text { A } & \$ 114,500 & 29 \\\hline \text { B } & \$ 149,900 & 16 \\\hline \text { C } & \$ 154,700 & 59 \\\hline \text { D } & \$ 159,900 & 42 \\\hline \text { E } & \$ 160,000 & 72 \\\hline \text { F } & \$ 165,900 & 45 \\\hline \text { G } & \$ 169,700 & 12 \\\hline \text { H } & \$ 171,900 & 39 \\\hline \text { I } & \$ 175,000 & 81 \\\hline \text { J } & \$ 289,900 & 121 \\\hline\end{array}$$ a. Find a 90% confidence interval for the mean number of days on the market for all
houses listed at $150,000.
b. Suppose a house has just been listed at $150,000. Find a 90% prediction interval for the
number of days the house will be on the market before it sells.

Question 42

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.

$\begin{array}{cccccc}\text { Horse } & \text { Gestation } & \text { Life } & \text { Horse } & \text { Gestation } & \text { Life } \\& \text { period } & \text { Length } & & \text { period } & \text { Length } \\& x \text { (days) } & y \text { (years) } & & x \text { (days) } & y \text { (years) } \\1 & 416 & 24 & 5 & 356 & 22 \\2 & 279 & 25.5 & 6 & 403 & 23.5 \\3 & 298 & 20 & 7 & 265 & 21 \\4 & 307 & 21.5 & & &\end{array}$

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 $\hat { y } = 18.89 + .01087 x$ .

Question 43

Plot the line . Then give the slope and y-intercept of the line.

Question 44

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.

$\begin{array}{lllll}\text { Horse}&\text { Gestation}&\text { Life }&\text { Horse }&\text { Gestation }&\text { Life }\\&\text { period }&\text { Length }&&\text { period } &\text { Length }\\&x \text { (days) } &y \text { (years) } && x \text { (days) } &y \text { (years) }\\1&416 & 24 & 5 & 356 & 22 \\2&279 & 25.5 & 6 & 403 & 23.5 \\3&298 & 20 & 7 & 265 & 21 \\4&307 & 21.5 & & &\end{array}$

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 $\sigma$ .

Question 45

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:

$\begin{array}{cc}\hline \text { Number of Grunts } & \text { Age (days) } \\\hline 83 & 118 \\61 & 134 \\32 & 148 \\37 & 153 \\56 & 160 \\33 & 167 \\55 & 176 \\10 & 182 \\13 & 188 \\\hline\end{array}$

Find and interpret the value of $r^{2}$

Question 46

Consider the following pairs of measurements:

$\begin{array}{|c|c|c|c|c|c|}\hline\text { x} & 5 & 8 & 3 & 4 & 9 \\\hline\text { y } & 6.2 & 3.4 & 7.5 & 8.1 & 3.2 \\\hline\end{array}$

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?

Question 47

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

$\begin{array}{l|c|c|c|c}\text { PREDICTOR } & & & & \\\text { VARIABLES } & \text { COEFFICIENT } & \text { STD ERROR } & \text { STUDENT'S T } & \text { P } \\\hline \text { CONSTANT } & 18.7171 & 0.63708 & 29.38 & 0.0000 \\\text { MAX } & -0.08365 & 0.00491 & -17.05 & 0.0000\end{array}$


$\begin{array}{llll}\text { R-SQUARED } & 0.6960 & \text { RESID. MEAN SQUARE (MSE) } & 1.28695 \\\text { ADJUSTED R-SQUARED } & 0.6937 & \text { STANDARD DEVIATION } & 1.13444\end{array}$

$\begin{array}{l|r|c|c|c|c}\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \mathrm{F} & \mathrm{P} \\\hline \text { REGRESSION } & 1 & 374.285 & 374.285 & 290.83 & 0.0000 \\\text { RESIDUAL } & 127 & 163.443 & 1.28695 & & \\\text { TOTAL } & 128 & 537.728 & & &\end{array}$


CASES INCLUDED 129 $\quad$ MISSING CASES 0

$\text { Find and interoret the estimate } \hat{\beta}_{1} \text { in the printout above. }$

Question 48

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 $\beta _ { 1 }$ .

Question 49

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:

$\begin{array}{cc}\hline \text {Number of Grunts }&\text { Age (days) }\\\hline 87 & 122 \\65 & 138 \\36 & 152 \\41 & 157 \\60 & 164 \\37 & 171 \\59 & 180 \\14 & 186 \\17 & 192 \\\hline\end{array}$

Find and interpret the value of r.

Question 50

Suppose you fit a least squares line to 22 data points and the calculated value of SSE is .678.

Question 51

Plot the line $y = 3 x$ . Then give the slope and y-intercept of the line.

Question 52

Plot the line $y = 1.5 + .5 x$ . Then give the slope and y-intercept of the line.

Question 53

a. Complete the table.
$\begin{array} { | l | c | c | c | c | } \hline & x _ { i } & y _ { i } & x _ { i } ^ { 2 } & x _ { i } y _ { i } \\\hline & 2 & 3 & & \\\hline & 5 & 2 & & \\\hline & 3 & 4 & & \\\hline & 8 & 0 & & \\\hline \text { Totals } & \Sigma x _ { i } = & \Sigma y _ { i } = & \Sigma x _ { i } ^ { 2 } = & \Sigma x _ { i } y _ { i } = \\\hline\end{array}$

b. Find $S S _ { x y } , S S _ { x x } , \beta _ { 1 } , \bar { x } , \bar { y }$ , and $\hat { \beta } _ { 0 }$ .
c. Write the equation of the least squares line.

Question 54

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

Question 55

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.

$\begin{array}{|c|c}\text { Months on Job } & \begin{array}{c}\text { Monthly Sales } \\y \text { (\$ thousands) }\end{array} \\\hline 2 & 2.4 \\4 & 7.0 \\8 & 11.3 \\12 & 15.0 \\1 & .8 \\5 & 3.7 \\9 & 12.0\end{array}$
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 $\bar { y } = 7.4571$ . Using SSE $= 12.435$ , find and interpret the coefficient of determination.

Question 56

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

$\begin{array}{l|c|c|c|c}\text { PREDICTOR } & & & & \\\text { VARIABLES } & \text { COEFFICIENT } & \text { STD ERROR } & \text { STUDENT'S T } & \text { P } \\\hline \text { CONSTANT } & 10.1639 & 1.77844 & 5.72 & 0.0000 \\\text { ORDERS } & 0.05836 & 0.00586 & 9.96 & 0.0000\end{array}$

$\begin{array}{lllr}\text { R-SQUARED } & 0.7229 & \text { RESID. MEAN SQUARE (MSE) } & 11.6175 \\\text { ADJUSTED R-SQUARED } & 0.7156 & \text { STANDARD DEVIATION } & 3.40844\end{array}$


$\begin{array}{l|r|r|r|c|c}\text { SOURCE } & \text { DF } &{\text { SS }} & \text { MS } & \mathrm{F} & \mathrm{P} \\\hline \text { REGRESSION } & 1 & 1151.55 & 1151.55 & 99.12 & 0.0000 \\\text { RESIDUAL } & 38 & 441.464 & 11.6175 & & \\\text { TOTAL } & 39 & 1593.01 & & &\end{array}$

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 $\alpha = .01$ .

Question 57

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.

$\begin{array}{c|c}\text { Months on Job } & \begin{array}{c}\text { Monthly Sales } \\y \text { (\$ thousands) }\end{array} \\\hline 2 & 2.4 \\4 & 7.0 \\8 & 11.3 \\12 & 15.0 \\1 & .8 \\5 & 3.7 \\9 & 12.0\end{array}$

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.

Question 58

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 $\alpha = .05$

Question 59

Question 60

Consider the following pairs of observations: Find and interpret the value of the coefficient of correlation.

Question 61

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.

$\begin{array} { l l } \hat { y } = 4.25 + 2.75 x , & S S x y = 5.15 , S S x x = 1.87 \\ & \text { SSyy } = 15.17 , S S E = 1.0075 \\ \\\text { Range of the } x \text {-values: } & 2.23 - 3.85\\\text {Range of the \(y\)-values:} &9.3 - 15.6 \end{array}$

Suppose a $95 \%$ prediction interval for $y$ when $x = 3.00$ is $( 16,21 )$ . Interpret the interval.

A) We are 95% confident that the starting salary of a Business School graduate with a GPA of 3.00 will fall between $16,000 and $21,000.
B) We are 95% confident that the starting salary of a Business School graduate will fall between $16,000 and $21,000.
C) We are 95% confident that the mean starting salary of all Business School graduates with GPAs of 3.00 will fall between $16,000 and $21,000.
D) We are 95% confident that the starting salary of a Business School graduate will increase between $16,000 and $21,000 for every 3-point increase in GPA.

Question 62

(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.
$$\begin{array} { l l l l l } & \text { School } & \text { GMAT } & \text { Acc. Rate } & \text { Salary } \\\hline\text { 1. } & \text { Harvard } & 644 & 15.0 \% & \$ 63,000 \\2 . & \text { Stanford } & 665 & 10.2 & 60,000 \\3 . & \text { Penn } & 644 & 19.4 & 55,000 \\4 . & \text { Northwestern } & 640 & 22.6 & 54,000 \\5 . & \text { MIT } & 650 & 21.3 & 57,000 \\6 . & \text { Chicago } & 632 & 30.0 & 55,269 \\7 . & \text { Duke } & 630 & 18.2 & 53,300 \\8 . & \text { Dartmouth } & 649 & 13.4 & 52,000 \\9 . & \text { Virginia } & 630 & 23.0 & 55,269 \\10 . & \text { Michigan } & 620 & 32.4 & 53.300 \\11 . & \text { Columbia } & 635 & 37.1 & 52,000 \\\text { 12. } & \text { Cornell } & 648 & 14.9 & 50,700 \\\text { 13. } & \text { CMU } & 630 & 31.2 & 52,050 \\14 . & \text { UNC } & 625 & 15.4 & 50,800 \\15 . & \text { Cal-Berkeley } & 634 & 24.7 & 50,000 \\16 . & \text { UCLA } & 640 & 20.7 & 51,494 \\\text { 17. } & \text { Texas } & 612 & 28.1 & 43,985 \\\text { 18. } & \text { Indiana } & 600 & 29.0 & 44,119 \\\text { 19. } & \text { NYU } & 610 & 35.0 & 53,161 \\\text { 20. } & \text { Purdue } & 595 & 26.8 & 43,500 \\21 . & \text { USC } & 610 & 31.9 & 49,080 \\\text { 22. } & \text { Pittsburgh } & 605 & 33.0 & 43,500 \\23 . & \text { Georgetown } & 617 & 31.7 & 45,156 \\24 . & \text { Maryland } & 593 & 28.1 & 42,925 \\25 . & \text { Rochester } & 605 & 35.9 & 44,499\end{array}$$
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 $t = 6.67$ .

A) There is evidence (at $\alpha = .05$ ) of at least a positive linear relationship between SALARY and GMAT.
B) We estimate SALARY to increase $\$ 6.67$ for every 1-point increase in GMAT.
C) There is evidence (at $\alpha = .05$ ) to indicate that $\beta _ { 1 } = 0$ .
D) Only $6.67 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight-line model.

Question 63

A realtor collected the following data for a random sample of ten homes that recently sold in her area.

$$\begin{array} { | c | c | c | } \hline \text { House } & \text { Asking Price } & \text { Days on Market } \\\hline \text { A } & \$ 114,500 & 29 \\\hline \text { B } & \$ 149,900 & 16 \\\hline \text { C } & \$ 154,700 & 59 \\\hline \text { D } & \$ 159,900 & 42 \\\hline \text { E } & \$ 160,000 & 72 \\\hline \text { F } & \$ 165,900 & 45 \\\hline \text { G } & \$ 169,700 & 12 \\\hline\end{array}$$ $$\begin{array} { | c | c | c | } \hline \mathrm { G } & \$ 169,700 & 12 \\\hline \mathrm { H } & \$ 171,900 & 39 \\\hline \mathrm { I } & \$ 175,000 & 81 \\\hline \mathrm { J } & \$ 289,900 & 121 \\\hline\end{array}$$
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, $x \text {. Use } \alpha = .05$

Question 64

Probabilistic models are commonly used to estimate both the mean value of $y$ and a new individual value of $y$ for a particular value of $x$ .

Question 65

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.

Question 66

Question 67

Is there a relationship between the raises administrators at State 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 straight-line regression model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$$ Using the method of least squares, the faculty group obtained the following prediction equation: $$\hat { y } = 14,000 - 2,000 x$$ Interpret the estimated y-intercept of the line.

A) There is no practical interpretation, since rating of 0 is nonsensical and outside the range of the sample data.
B) For an administrator who receives a rating of zero, we estimate his or her raise to be $14,000.
C) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $14,000.
D) The base administrator raise at State University is $14,000.

Question 68

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

Question 69

Consider the following pairs of observations: $\begin{array}{|c|c|c|c|c|c|}\hline x & 2 & 3 & 5 & 5 & 6 \\\hline y & 1.3 & 1.6 & 2.1 & 2.2 & 2.7 \\\hline\end{array}$

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 $\alpha = .05$ .

Question 70

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: $$\sum y = 1,134 , \sum ^ { x } = 3.642 , \sum y ^ { 2 } = 93,110 , \sum ^ { 2 } = .948622 , \text { and } \sum x y = 295.54$$
Assume $\hat { \beta } 1 = 455.27$ and $\hat { \sigma } = 9.18$ . Conduct a test of hypothesis to determine if a positive linear relationship exists between team batting average and number of wins. Use $\alpha = .05$ .

Question 71

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.
$\begin{array}{llc}\hline \text { Firm } & \text { Estimated Market Model } & \text { Estimated Standard Error of } \beta 1 \\\hline \text { Company A } & y=.0010+1.40 x & .03 \\\text { Company B } & y=.0005-1.21 x & .06 \\\text { Company C } & y=.0010+1.62 x & 1.34 \\\text { Company D } & y=.0013+.76 x & .15 \\\hline\end{array}$

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 $x$ ?

A) Companies B and D only
B) Company D only
C) Companies B and C only
D) Company C only

Question 72

(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.
$$\begin{array} { l l l l l } & \text { School } & \text { GMAT } & \text { Acc. Rate } & \text { Salary } \\\hline\text { 1. } & \text { Harvard } & 644 & 15.0 \% & \$ 63,000 \\2 . & \text { Stanford } & 665 & 10.2 & 60,000 \\3 . & \text { Penn } & 644 & 19.4 & 55,000 \\4 . & \text { Northwestern } & 640 & 22.6 & 54,000 \\5 . & \text { MIT } & 650 & 21.3 & 57,000 \\6 . & \text { Chicago } & 632 & 30.0 & 55,269 \\7 . & \text { Duke } & 630 & 18.2 & 53,300 \\8 . & \text { Dartmouth } & 649 & 13.4 & 52,000 \\9 . & \text { Virginia } & 630 & 23.0 & 55,269 \\10 . & \text { Michigan } & 620 & 32.4 & 53.300 \\11 . & \text { Columbia } & 635 & 37.1 & 52,000 \\\text { 12. } & \text { Cornell } & 648 & 14.9 & 50,700 \\\text { 13. } & \text { CMU } & 630 & 31.2 & 52,050 \\14 . & \text { UNC } & 625 & 15.4 & 50,800 \\15 . & \text { Cal-Berkeley } & 634 & 24.7 & 50,000 \\16 . & \text { UCLA } & 640 & 20.7 & 51,494 \\\text { 17. } & \text { Texas } & 612 & 28.1 & 43,985 \\\text { 18. } & \text { Indiana } & 600 & 29.0 & 44,119 \\\text { 19. } & \text { NYU } & 610 & 35.0 & 53,161 \\\text { 20. } & \text { Purdue } & 595 & 26.8 & 43,500 \\21 . & \text { USC } & 610 & 31.9 & 49,080 \\\text { 22. } & \text { Pittsburgh } & 605 & 33.0 & 43,500 \\23 . & \text { Georgetown } & 617 & 31.7 & 45,156 \\24 . & \text { Maryland } & 593 & 28.1 & 42,925 \\25 . & \text { Rochester } & 605 & 35.9 & 44,499\end{array}$$
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 $\hat { \beta } _ { 0 } = - 92040$ .

A) The value has no practical interpretation since a GMAT of 0 is nonsensical and outside the range of the sample data.
B) We expect to predict SALARY to within $2 ( 92040 ) = \$ 184,080$ of its true value using GMAT in a straight-line model.
C) We estimate the base SALARY of graduates of a top business school to be $- \$ 92,040$ .
D) We estimate SALARY to decrease $\$ 92,040$ for every 1-point increase in GMAT.

Question 73

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

Question 74

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:

$\begin{array}{cc}\hline \text { Number of Grunts } & \text { Age (days) } \\\hline 102 & 130 \\80 & 146 \\51 & 160 \\56 & 165 \\75 & 172 \\52 & 179 \\74 & 188 \\29 & 194 \\34 & 200 \\\hline\end{array}$

a. Find $S S E , s ^ { 2 }$ , and $s$ .
b. Interpret the value of $s$ .

Question 75

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

A)


B)


C)


D)

Question 76

Graph the line that passes through the given points.

- $(-6,0) \text { and }(1,-1)$

A)


B)


C)


D)

Question 77

Is there a relationship between the raises administrators at State 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 straight-line regression model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$$ Using the method of least squares, the faculty group obtained the following prediction equation: $$\hat { y } = 14,000 - 2,000 x$$ Interpret the estimated slope of the line.

A) For an administrator with a rating of 1.0, we estimate his/her raise to be $\$ 2,000$ .
B) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to increase $\$ 2,000$ .
C) For a 1-point increase in an administrator's rating, we estimate the administrator's raise to decrease $\$ 2,000$ .
D) For a $\$ 1$ increase in an administrator's raise, we estimate the administrator's rating to decrease 2,000 points.

Question 78

(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.
$$\begin{array} { l l l l l } & \text { School } & \text { GMAT } & \text { Acc. Rate } & \text { Salary } \\\hline\text { 1. } & \text { Harvard } & 644 & 15.0 \% & \$ 63,000 \\2 . & \text { Stanford } & 665 & 10.2 & 60,000 \\3 . & \text { Penn } & 644 & 19.4 & 55,000 \\4 . & \text { Northwestern } & 640 & 22.6 & 54,000 \\5 . & \text { MIT } & 650 & 21.3 & 57,000 \\6 . & \text { Chicago } & 632 & 30.0 & 55,269 \\7 . & \text { Duke } & 630 & 18.2 & 53,300 \\8 . & \text { Dartmouth } & 649 & 13.4 & 52,000 \\9 . & \text { Virginia } & 630 & 23.0 & 55,269 \\10 . & \text { Michigan } & 620 & 32.4 & 53.300 \\11 . & \text { Columbia } & 635 & 37.1 & 52,000 \\\text { 12. } & \text { Cornell } & 648 & 14.9 & 50,700 \\\text { 13. } & \text { CMU } & 630 & 31.2 & 52,050 \\14 . & \text { UNC } & 625 & 15.4 & 50,800 \\15 . & \text { Cal-Berkeley } & 634 & 24.7 & 50,000 \\16 . & \text { UCLA } & 640 & 20.7 & 51,494 \\\text { 17. } & \text { Texas } & 612 & 28.1 & 43,985 \\\text { 18. } & \text { Indiana } & 600 & 29.0 & 44,119 \\\text { 19. } & \text { NYU } & 610 & 35.0 & 53,161 \\\text { 20. } & \text { Purdue } & 595 & 26.8 & 43,500 \\21 . & \text { USC } & 610 & 31.9 & 49,080 \\\text { 22. } & \text { Pittsburgh } & 605 & 33.0 & 43,500 \\23 . & \text { Georgetown } & 617 & 31.7 & 45,156 \\24 . & \text { Maryland } & 593 & 28.1 & 42,925 \\25 . & \text { Rochester } & 605 & 35.9 & 44,499\end{array}$$
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 $\mathrm { s } = 3213$ .

A) We expect to predict SALARY to within $2 ( 3213 ) = \$ 6,426$ of its true value using GMAT in a straight-line model.
B) Our predicted value of SALARY will equal $2 ( 3213 ) = \$ 6,426$ for any value of GMAT.
C) We estimate SALARY to increase $\$ 3,213$ for every 1-point increase in GMAT.
D) We expect the predicted SALARY to deviate from actual SALARY by at least $2 ( 3213 ) = \$ 6,426$ using GMAT in a straight-line model.

Question 79

Construct a 90% confidence interval for $\beta _ { 1 } \text { when } \hat { \beta } 1 = 49 , s = 4 , \mathrm { SS } _ { \mathrm { Xx } } = 55 , \text { and } n = 15 \text {. }$

Question 80

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 .

A) There is insufficient evidence (at $\alpha = .05$ ) to conclude that service charge $( y )$ is positively linearly related to sales revenue $( x )$ .
B) Sales revenue $( x )$ is a poor predictor of service charge $( y )$ .
C) There is sufficient evidence (at $\alpha = .05$ ) to conclude that service charge $( y )$ is positively linearly related to sales revenue $( x )$ .
D) For every $\$ 1$ million increase in sales revenue $( x )$ , we expect a service charge $( y )$ to increase $.064.