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

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. 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. C

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) $S S E = 4512.024 ; s ^ { 2 } = 139.843 ; s = 11.826$ 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) $S S E = 4512.024 ; s ^ { 2 } = 139.843 ; s = 11.826$

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.

a. @#LAT-DLM& b. @#LAT-DLM& c. We would expect approximately

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

Consider the following pairs of observations: \[\begin{array} { | c | c | c | c | c | c | } \hline x & 2 & 3 & 5 & 5 & 6 \\ \hline y & 1.3 & 1.6 & 2.1 & 2.2 & 2.7 \\ \hline \end{array}\] Find and interpret the value of the coefficient of determination.

@#LAT-DLM& @#LAT-DLM& of the sample vari

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. 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.

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 $\alpha = 0.05$. 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 $\alpha = 0.05$.

Exam 11: Simple Linear Regression

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.

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Question 1

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$\begin{array} { l } \beta _ { 1 } = \frac { S S _ { x y } } { S S _ { x x } } = \frac { 25 } { 10.5 } \approx 2.3810 \\\\\hat { \beta } _ { 0 } = \bar { y } - \beta _ { 1 } x = 31 - 2.3810 ( 2.75 ) = 24.4523\end{array}$
The least squares prediction equation is $\hat { y } = 24.4523 + 2.3810 \mathrm { x }$

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 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 $\quad 0.5972 \quad$ 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.

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Question 2

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B

A study of the top 75 MBA programs attempted to predict the average starting salary (in $1000's) of graduates of the program based on the amount of tuition (in $1000's) charged by the program. The results of a simple linear regression analysis are shown below: Least Squares Linear Regression of Salary Predictor Variables Coefficient Std Error T P Constant 18.1849 10.3336 1.76 0.0826 Size 1.47494 0.14017 10.52 0.0000 R-Squared 0.6027 Resid. Mean Square (MSE) 532.986 Adjusted R-Squared 0.5972 Standard Deviation 23.0865 Interpret the estimated slope of the regression line.

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C

Graph the line that passes through the given points. - $( 3 , - 5 ) \text { and } ( 9,6 )$ $Graph the line that passes through the given points. - ( 3 , - 5 ) \text { and } ( 9,6 )$

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Question 4

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

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Question 5

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

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Question 6

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

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Question 7

An academic advisor wants to predict the typical starting salary of a graduate at a top business school using the GMAT score of the school as a predictor variable. A simple linear regression of SALARY versus GMAT using 25 data points is shown below. $\hat { \beta } _ { 0 } = - 92040 \hat { \beta } 1 = 228 \mathrm {~s} = 3213 \mathrm { df } = 23 \quad t = 6.67$ Set up the null and alternative hypotheses for testing whether a linear relationship exists between SALARY and GMAT.

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Question 8

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

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Question 9

In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and and the age of the warthog (in days) are listed below: Number of Grunts Age (days) 83 118 61 134 32 148 37 153 56 160 33 167 55 176 10 182 13 188 Find and interpret the value of $r^{2}$

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Question 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 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.

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Question 11

Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

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Question 12

Consider the following pairs of observations: x 2 3 5 5 6 y 1.3 1.6 2.1 2.2 2.7 Find and interpret the value of the coefficient of determination.

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Question 13

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 $\alpha = .05$ .

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Question 14

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 coefficient of determination for the regression model.

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Question 15

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.

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Question 16

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

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Question 17

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 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?

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Question 18

Consider the following pairs of measurements: x 1 3 4 6 7 y 3 6 8 12 13 a. Construct a scattergram for the data. b. What does the scattergram suggest about the relationship between $x$ and $y$ ? c. Find the least squares estimates of $\beta _ { 0 }$ and $\beta _ { 1 }$ . d. Plot the least squares line on your scattergram. Does the line appear to fit the data well?

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Question 19

Suppose you fit a least squares line to 22 data points and the calculated value of SSE is .678. a. Find $s ^ { 2 }$ , the estimator of $\sigma ^ { 2 }$ . b. Find $s$ , the estimator of $\sigma$ . c. What is the largest deviation you might expect between any one of the 22 points and the least squares line?

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Question 20

Graph the line that passes through the given points. - $( 3 , - 5 ) \text { and } ( 9,6 )$ $Graph the line that passes through the given points. - ( 3 , - 5 ) \text { and } ( 9,6 )$

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

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

Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

Consider the following pairs of observations: x 2 3 5 5 6 y 1.3 1.6 2.1 2.2 2.7 Find and interpret the value of the coefficient of determination.

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

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

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Exam 11: Simple Linear Regression

Graph the line that passes through the given points. - (3,−5) and (9,6)( 3 , - 5 ) \text { and } ( 9,6 )(3,−5) and (9,6)

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

Construct a 95% confidence interval for β1 when β^1=49,s=4,SSXx=55, and n=15. \beta _ { 1 } \text { when } \hat { \beta } 1 = 49 , s = 4 , \mathrm { SS } _ { \mathrm { Xx } } = 55 \text {, and } n = 15 \text {. }β1​ when β^​1=49,s=4,SSXx​=55, and n=15.

Consider the data set shown below. Find the 95% confidence interval for the slope of the regression line. 0 3 2 3 8 10 11 -2 0 2 4 6 8 10

Consider the following pairs of observations: x 2 3 5 5 6 y 1.3 1.6 2.1 2.2 2.7 Find and interpret the value of the coefficient of determination.

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

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Random Variables and Probability Distributions

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single Sample: 355 Tests of Hypotheses

Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Design of Experiments and Analysis of Variance

Categorical Data Analysis

Multiple Regression and Model Building

Methods for Quality Improvement: Statistical Process Control Available on CD

Time Series: Descriptive Analyses, Models, and Forecasting Available on CD

Nonparametric Statistics Available on CD

Filters

Graph the line that passes through the given points. - $( 3 , - 5 ) \text { and } ( 9,6 )$ $Graph the line that passes through the given points. - ( 3 , - 5 ) \text { and } ( 9,6 )$

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

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

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