Question 1

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  Interpret the estimated y-intercept of the regression line.

Accepted Answer

A)  When a diamond is 0 carats in size, we estimate the price of the diamond to be $11,598.90. 
B)  When a diamond is 0 carats in size, we estimate the price of the diamond to be $2298.36. 
C)  When a diamond is 11598.9 carats in size, we estimate the price of the diamond to be $2298.36. 
D)  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.

Question 2

A manufacturer of boiler drums wants to use regression to predict the number of man-hours needed to erect drums in the future. The manufacturer collected a random sample of 35 boilers and measured the following two variables: MANHRS: $\quad y =$ Number of man-hours required to erect the drum
PRESSURE: $\quad x _ { 1 } =$ Boiler design pressure (pounds per square inch, i.e., $\mathrm { psi }$ )
The simple linear model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to the data. A printout for the analysis appears below:

UNWEIGHTED LEAST SQUARES LINEAR REGRESSION OF MANHRS
$\begin{array}{c|c|c|c|c}
\text { PREDICTOR } & & & & \
\text { VARIABLES } & \text { COEFFICIENT } & \text { STD ERROR } & \text { STUDENT'S T } & \text { P } \
\hline \text { CONSTANT } & 1.88059 & 0.58380 & 3.22 & 0.0028 \
\text { PRESSURE } & 0.00321 & 0.00163 & 2.17 & 0.0300
\end{array}$

$\begin{array}{l|r|c|c|c|c}
\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \mathrm{F} & \mathrm{P} \
\hline \text { REGRESSION } & 1 & 111.008 & 111.008 & 5.19 & 0.0300 \
\text { RESIDUAL } & 34 & 144.656 & 4.25160 & & \
\text { TOTAL } & 35 & 255.665 & & &
\end{array}$
 Fill in the blank. At ? =.01, there is ____________ between man-hours and pressure.

Accepted Answer

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

Question 3

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

Accepted Answer

The answer of Plot the line y = 3x. Then...

Question 4

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

Accepted Answer

The answer of A large national bank charges local companies...

Question 5

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 $$\begin{array} { l c c c c c } 
\text { Predictor } & & & & \
\text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } & \
\text { Constant } & - 2298.36 & 158.531 & - 14.50 & 0.0000 & \
\text { Size } & 11598.9 & 230.111 & 50.41 & 0.0000 & \
& & & & & \
\text { R-Squared } & 0.8925 & \text { Resid. Mean Square (MSE) } & 1248950 \
\text { Adjusted R-Squared } & 0.8922 & { \text { Standard Deviation } } & 1117.56
\end{array}$$ Interpret the coefficient of determination for the regression model.

Accepted Answer

A)  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. 
B)  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. 
C)  We expect most of the sampled diamond prices to fall within $2235.12 of their least squares predicted values. 
D)  For every 1-carat increase in the size of a diamond, we estimate that the price of the diamond will increase by $1117.56.
11.6 Using the Model for Estimation and Prediction
1 Calculate and Compare Confidence Intervals for Mean of y

Question 6

Set up the null and alternative hypotheses for testing whether a positive linear relationship exists between SALARY and GMAT in the situation above. A) $H _ { 0 } : \beta _ { 1 } = 0$ vs. $H _ { a } : \beta _ { 1 } > 0$
B) $H _ { 0 } : \beta _ { 1 } = 0$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } \neq 0$
C) $H _ { 0 } : \beta _ { 1 } = 228$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } > 228$
D) $H _ { 0 } : \beta _ { 1 } > 0$ vs. $H _ { \mathrm { a } } : \beta _ { 1 } < 0$

Accepted Answer

The answer of Set up the null and alternative hypotheses...

Question 7

Consider the data set shown below. Find the coefficient of determination for the simple linear regression model. $$\begin{array} { c | c | c | c | c | c | c | c } 
\mathrm { y } & 0 & 3 & 2 & 3 & 8 & 10 & 11 \
\hline \mathrm { x } & - 2 & 0 & 2 & 4 & 6 & 8 & 10
\end{array}$$

Accepted Answer

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

Question 8

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}{l}
\begin{array} { c c } 
\hline \text { Number of Grunts } & \text { Age (days) } \
\hline 93 & 128 \
71 & 144 \
42 & 158 \
47 & 163 \
66 & 170 \
43 & 177 \
65 & 186 \
20 & 192 \
23 & 198 \
\hline
\end{array}\

\text { Find and interpret the value of } r ^ { 2 } \text {. }
\end{array}$$

Accepted Answer

The answer of In a study of feeding behavior, zoologists...

Question 9

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 bank's 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 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.
$$\hat { y } = 2,700 + 20 x$$
Interpret the estimate of $\beta _ { 0 }$, the $y$-intercept of the line.

Accepted Answer

A)  There is no practical interpretation since a sales revenue of $0 is a nonsensical value. 
B)  All companies will be charged at least $2,700 by the bank. 
C)  About 95% of the observed service charges fall within $2,700 of the least squares line. 
D)  For every $1 million increase in sales revenue, we expect a service charge to increase $2,700.

Question 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. $$\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 s = 3213 r ^ { 2 } = .66 r = .81 \mathrm { df } = 23 \quad t = 6.67$$ Give a practical interpretation of r = .81.

Accepted Answer

A)  There appears to be a positive correlation between SALARY and GMAT. 
B)  We estimate SALARY to increase 81% for every 1-point increase in GMAT. 
C)  81% of the sample variation in SALARY can be explained by using GMAT in a straight -line model. 
D)  We can predict SALARY correctly 81% of the time using GMAT in a straight-line model.

Question 11

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

Accepted Answer

A) True 
 B)False

Question 12

For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid?

Accepted Answer

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 13

For the situation above, give a practical interpretation of t = 6.67.

Accepted Answer

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

Question 14

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

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.

Accepted Answer

The answer of a. Complete the table.
\(\begin{array}{|l|c|c|c|c|}
\hline & x_{i} &...

Question 15

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.

Accepted Answer

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

Question 16

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 restults were obtained:
$$\hat { y } = 74.80 + 21.66 x$$ Give a practical interpretation of the estimate of the slope of the least squares line.

Accepted Answer

A)  For each additional room in the house, we estimate the appraised value to increase $21,660. 
B)  For each additional room in the house, we estimate the appraised value to increase $74,800. 
C)  For each additional dollar of appraised value, we estimate the number of rooms in the house to increase by 21.66. 
D)  For a house with 0 rooms, we estimate the appraised value to be $74,800.

Question 17

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: $$\sum y = 1,134 , \sum x = 3.642 , \sum y ^ { 2 } = 93,110 , \sum x^ { 2 } = .948622 , \text { and } \sum x y = 295.54$$
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 $\alpha = .05$.

Accepted Answer

The answer of Is the number of games won by...

Question 18

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

Accepted Answer

The answer of Plot the line y = 1.5 +...

Question 19

(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. \[\begin{array} { l l l l l } 
& \text { School } & \text { GMAT } & \text { Acc. Rate } & \text { Salary } \
\hline \text { 1. } & \text { Harvard } & 644 & 15.0 \% & \$ 63,000 \
\text { 2. } & \text { Stanford } & 665 & 10.2 & 60,000 \
\text { 3. } & \text { Penn } & 644 & 19.4 & 55,000 \
\text { 4. } & \text { Northwestern } & 640 & 22.6 & 54,000 \
\text { 5. } & \text { MIT } & 650 & 21.3 & 57,000 \
\text { 6. } & \text { Chicago } & 632 & 30.0 & 55,269 \
\text { 7. } & \text { Duke } & 630 & 18.2 & 53,300 \
\text { 8. } & \text { Dartmouth } & 649 & 13.4 & 52,000 \
\text { 9. } & \text { Virginia } & 630 & 23.0 & 55,269 \
\text { 10. } & \text { Michigan } & 620 & 32.4 & 53.300 \
\text { 11. } & \text { Columbia } & 635 & 37.1 & 52,000 \
\text { 12. } & \text { Cornell } & 648 & 14.9 & 50,700 \
\text { 13. } & \text { CMU } & 630 & 31.2 & 52,050 \
\text { 14. } & \text { UNC } & 625 & 15.4 & 50,800 \
\text { 15. } & \text { Cal-Berkeley } & 634 & 24.7 & 50,000 \
\text { 16. } & \text { UCLA } & 640 & 20.7 & 51,494 \
\text { 17. } & \text { Texas } & 612 & 28.1 & 43,985 \
\text { 18. } & \text { Indiana } & 600 & 29.0 & 44,119 \
\text { 19. } & \text { NYU } & 610 & 35.0 & 53,161 \
\text { 20. } & \text { Purdue } & 595 & 26.8 & 43,500 \
\text { 21. } & \text { USC } & 610 & 31.9 & 49,080 \
\text { 22. } & \text { Pittsburgh } & 605 & 33.0 & 43,500 \
\text { 23. } & \text { Georgetown } & 617 & 31.7 & 45,156 \
\text { 24. } & \text { Maryland } & 593 & 28.1 & 42,925 \
\text { 25. } & \text { Rochester } & 605 & 35.9 & 44,499
\end{array}\] 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 $r ^ { 2 } = .66$.
A) We expect to predict SALARY to within $2 ( \sqrt { .66 } )$ of its true value using GMAT in a straight-line model.
B) We estimate SALARY to increase $\$ .66$ for every 1 -point increase in GMAT.
C) $66 \%$ of the sample variation in SALARY can be explained by using GMAT in a straight -line model.
D) We can predict SALARY correctly $66 \%$ of the time using GMAT in a straight-line model.

Accepted Answer

The answer of (Situation P) Below are the results of...

Question 20

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

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 $\hat { y } = - .25 + 1.315 x$.

Accepted Answer

The answer of A company keeps extensive records on its...

Exam 11: Simple Linear Regression

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

Consider the data set shown below. Find the coefficient of determination for the simple linear regression model. $\begin{array} { c | c | c | c | c | c | c | c } \mathrm { y } & 0 & 3 & 2 & 3 & 8 & 10 & 11 \\\hline \mathrm { x } & - 2 & 0 & 2 & 4 & 6 & 8 & 10\end{array}$

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

For the situation above, which of the following is not an assumption required for the simple linear regression analysis to be valid?

For the situation above, give a practical interpretation of t = 6.67.

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