A study of the top 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 and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: \[\text { Least Squares Linear Regression of Salary }\] $\begin{array}{l} \text { Predictor }\\ \begin{array}{lccccc} \text { Variables } & \text { Coefficient } & \text { Std Error } & \text { T } & \text { P } \\ \text { Constant } & -687.851 & 165.406 & 4.16 & 0.0001 \\ \text { Tuition } & -11.3197 & 2.19724 & -5.15 & 0.0000 \\ \text { GMAT } & -0.96727 & 0.25535 & -3.79 & 0.0003 \\ \text { TxG } & 0.01850 & 0.00331 & 5.58 & 0.0000 \end{array} \end{array}$ $\begin{array}{lccc} \text { R-Squared } & 0.7816 & \text { Resid. Mean Square (MSE) } & 301.251 \\ \text { Adjusted R-Squared } & 0.7723 & \text { Standard Deviation } & 17.3566 \end{array}$ $\begin{array}{lllcccc} \text { Source } & \text { DF } & & \text { SS } & \text { MS } & \text { F } & \text { P } \\ \text { Regression } & 3 & & 76523.8 & 25510.9 & 84.68 & 0.0000 \\ \text { Residual } & & 71 & 21388.8 & 301.3 & \\ \text { Total } & & 74 & 97921.7 & & \end{array}$ Cases Included 75 Missing Cases 0 One of the t-test test statistics is shown on the printout to be value $t = 5.58$. Interpret this value.

A) There is sufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. B) There is insufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. C) There is sufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. D) There is insufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. A) There is sufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs. B) There is insufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. C) There is sufficient evidence, at $\alpha = 0.05$, to indicate that at least one of the variables proposed in the interaction model is useful at predicting the average starting salary of graduates of MBA programs. D) There is insufficient evidence, at $\alpha = 0.05$, to indicate that the interaction between average tuition and average GMAT score is a useful predictor of the average starting salary of graduates of MBA programs.

Exam 12: Multiple Regression and Model Building

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of x far outside the range of the sampled values of x.

(True/False)

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Question 121

A first-order model may include terms for both quantitative and qualitative independent variables.

(True/False)

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Question 122

Consider the data given in the table below. 1 4 2 6 2 5 3 7 4 7 4 6 5 4 5 5 6 3 a. Plot the data on a scattergram. Does a quadratic model seem to be a good fit for the data? Explain. b. Use the method of least squares to find a quadratic prediction equation. c. Graph the prediction equation on your scattergram.

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Question 123

A study of the top 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 and the average GMAT score of the program's students. The results of a regression analysis based on a sample of 75 MBA programs is shown below: $\text { Least Squares Linear Regression of Salary }$ Predictor Variables Coefficient Std Error T P Constant -687.851 165.406 4.16 0.0001 Tuition -11.3197 2.19724 -5.15 0.0000 GMAT -0.96727 0.25535 -3.79 0.0003 TxG 0.01850 0.00331 5.58 0.0000 R-Squared 0.7816 Resid. Mean Square (MSE) 301.251 Adjusted R-Squared 0.7723 Standard Deviation 17.3566 Source DF SS MS F P Regression 3 76523.8 25510.9 84.68 0.0000 Residual 71 21388.8 301.3 Total 74 97921.7 Cases Included 75 Missing Cases 0 One of the t-test test statistics is shown on the printout to be value $t = 5.58$ . Interpret this value.

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Question 124

Retail price data for $n = 60$ hard disk drives were recently reported in a computer magazine. Three variables were recorded for each hard disk drive: $y =$ Retail PRICE (measured in dollars) $x _ { 1 } =$ Microprocessor SPEED (measured in megahertz) (Values in sample range from 10 to 40 ) $x _ { 2 } = \mathrm { CHIP }$ size (measured in computer processing units) (Values in sample range from 286 to 486 ) A first-order regression model was fit to the data. Part of the printout follows: $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ Parameter Estimates $\quad$ $\quad$ $\quad$ PARAMETER STANDARD $\quad$ $\quad$ T FOR 0: VARIABLE DF ESTIMATE ERROR PARAMETER $= 0$ PROB $> | T |$ INTERCEPT 1 -373.526392 1258.1243396 -0.297 0.7676 SPEED 1 104.838940 22.36298195 4.688 0.0001 CHIP 1 3.571850 3.89422935 0.917 0.3629 Identify and interpret the estimate for the SPEED $\beta$ -coefficient, $\hat { \beta } _ { 1 }$ .

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Question 125

The table below shows data for $n = 20$ observations. 1 2 18 3 8 23 5 10 15 2 7 31 6 12 24 4 9 28 5 11 17 2 7 19 3 8 30 7 10 28 5 8 14 3 6 32 7 11 17 2 8 24 5 10 26 6 11 27 6 11 21 3 6 31 7 13 19 2 8 25 5 10 a. Use a first-order regression model to find a least squares prediction equation for the model. b. Find a $95 \%$ confidence interval for the coefficient of $x _ { 1 }$ in your model. Interpret the result. c. Find a $95 \%$ confidence interval for the coefficient of $x _ { 2 }$ in your model. Interpret the result. d. Find $R ^ { 2 }$ and $R _ { a } 2$ and interpret these values. e. Test the null hypothesis $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$ against the alternative hypothesis $H _ { \mathrm { a } } :$ at least one $\beta _ { i } \neq 0$ . Use $\alpha = .05$ . Interpret the result.

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Question 126

The confidence interval for the mean E(y) is narrower that the prediction interval for y.

(True/False)

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Question 127

During its manufacture, a product is subjected to four different tests in sequential order. An efficiency expert claims that the fourth (and last) test is unnecessary since its results can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test4 score $( y )$ , as a function of Test1 score $\left( x _ { 1 } \right)$ , Test 2 score $\left( x _ { 2 } \right)$ , and Test3 score ( $\left. x _ { 3 } \right)$ . [Note: All test scores range from 200 to 800 , with higher scores indicative of a higher quality product.] Consider the model: $E ( y ) = \beta _ { 1 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$ The global $F$ statistic is used to test the null hypothesis, $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$ . Describe this hypothesis in words.

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Question 128

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }$ indicates downward concavity.

(True/False)

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Question 129

The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables.

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Question 130

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$ was used to relate $E ( y )$ to a single qualitative variable. How many levels does the qualitative variable have?

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Question 131

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of x far outside the range of the sampled values of x.

A first-order model may include terms for both quantitative and qualitative independent variables.

The confidence interval for the mean E(y) is narrower that the prediction interval for y.

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }$ indicates downward concavity.

The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables.

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$ was used to relate $E ( y )$ to a single qualitative variable. How many levels does the qualitative variable have?

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

Simple Linear Regression

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

Exam 12: Multiple Regression and Model Building

In the presence of multicollinearity, the predicted values of y^\hat { y }y^​ are actually quite good for values of x far outside the range of the sampled values of x.

A first-order model may include terms for both quantitative and qualitative independent variables.

The confidence interval for the mean E(y) is narrower that the prediction interval for y.

In the quadratic model E(y)=β0+β1x+β2x2, a negative value of β1E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }E(y)=β0​+β1​x+β2​x2, a negative value of β1​ indicates downward concavity.

The complete second-order model with two quantitative independent variables does not allow for interaction between the two independent variables.

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

Simple Linear Regression

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

In the presence of multicollinearity, the predicted values of $\hat { y }$ are actually quite good for values of x far outside the range of the sampled values of x.

In the quadratic model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 } , \text { a negative value of } \beta _ { 1 }$ indicates downward concavity.