Question 1

It is safe to conduct t-tests on the individual β parameters in a first-order linear model in order to determine which independent variables are useful for predicting y and which are not.

Accepted Answer

Conducting t-tests on individual β parameters in a linear model does not take into account potential interactions between variables and may lead to incorrect conclusions. It is recommended to use other methods, such as stepwise regression or analysis of variance, to determine which variables are significant predictors of the outcome.

Question 2

A qualitative variable whose outcomes are assigned numerical values is called a coded variable.

Accepted Answer

A coded variable is indeed a qualitative variable for which numerical values are assigned to represent categories, making it easier to analyze statistically.

Question 3

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: Least Squares Linear Regression of Salary

Identify the test statistic that should be used to test to determine if the amount of tuition charged by a program is a useful predictor of the average starting salary of the graduates of the program.
A) $t = - 3.94$
B) $\mathrm { t } = 4.36$
C) $t = 5.15$
D) $t = 20.67$

Accepted Answer

C

Question 4

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 first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout.
SOURCE DF SS $\quad$ MS $\quad$ FVALUE $\quad$ PROB $>$ F
 
Suppose the $95 \%$ confidence interval for $\beta _ { 3 }$ is $( .15 , .47 )$. Which of the following statements is incorrect?
A) At $\alpha = .05$, there is insufficient evidence to reject $H _ { 0 } : \beta _ { 3 } = 0$ in favor of $H _ { a } : \beta _ { 3 } \neq 0$.
B) We are $95 \%$ confident that the increase in Test 4 score for every 1 -point increase in Test3 score falls between 15 and $.47$, holding Test1 and Test 2 fixed.
C) We are $95 \%$ confident that the Test 3 is a useful linear predictor of Test 4 score, holding Test1 and Test 2 fixed.
D) We are $95 \%$ confident that the estimated slope for the Test4-Test 3 line falls between. 15 and .47 holding Test 1 and Test 2 fixed.

Accepted Answer

The answer of During its manufacture, a product is subjected...

Question 5

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: Least Squares Linear Regression of Salary  Interpret the coefficient for the tuition variable shown on the printout.
A) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will decrease by $203,402, holding the GMAT score constant.
B) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $394.12, holding the GMAT score constant
C) For every $1000 increase in the tuition charged by the MBA program, we estimate that the average starting salary will increase by $920.12, holding the GMAT score constant
D) For every $1000 increase in the average starting salary, we estimate that the tuition charged by the MBA program will increase by $920.12.

Accepted Answer

The answer of A study of the top MBA programs...

Question 6

Why is the random error term ε added to a multiple regression model? 12.2 Estimating and Making Inferences about the β Parameters 1 Write First-Order Model

Accepted Answer

The answer of Why is the random error term ε...

Question 7

As part of a study at a large university, data were collected on n = 224 freshmen computer science (CS) majors in a particular year. The researchers were interested in modeling y, a student's grade point average (GPA) after three semesters, as a function of the following independent variables (recorded at the time the students enrolled in the university): $x _ { 1 } =$ average high school grade in mathematics (HSM)
$x _ { 2 } =$ average high school grade in science (HSS)
$x _ { 3 } =$ average high school grade in English (HSE)
$x _ { 4 } =$ SAT mathematics score (SATM)
$x _ { 5 } =$ SAT verbal score (SATV)

A first-order model was fit to data.

A $95 \%$ confidence interval for $\beta _ { 1 }$ is $( .06 , .22 )$. Interpret this result.

A) We are $95 \%$ confident that a CS freshman's GPA increases by an amount between .06 and .22 for every 1 -point increase in average H5 math grade, holding $x _ { 2 } - x _ { 5 }$ constant.
B) $95 \%$ of the GPAs fall within .06 to $.22$ of their true values.
C) We are $95 \%$ confident that a CS freshman's HS math grade increases by an amount between .06 and .22 for every 1-point increase in GPA, holding $x _ { 2 } - x _ { 5 }$ constant.
D) We are $95 \%$ confident that the mean GPA of all CS freshmen after three semesters falls between .06 and $.22$.

Accepted Answer

The answer of As part of a study at a...

Question 8

For a multiple regression model, we assume that the mean of the probability distribution of the random error is 0.

Accepted Answer

In a multiple regression model, we assume that the random error follows a normal distribution with mean 0 and constant variance. This assumption is important for accurate estimation of the model parameters and for making valid statistical inferences.

Question 9

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 restults can be predicted based on the first three tests. To test this claim, multiple regression will be used to model Test 4 score (y), as a function of Test1 score $\left( x _ { 1 } \right)$. Test 2 score $\left( x _ { 2 } \right)$, and Test 3 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 first-order model was fit to the data for each of 12 units sampled from the production line. The results are summarized in the printout.

A) $.33 \pm .19$
B) $33 \pm 105$
C). $.33 \pm .08$
D) $.33 \pm 4.04$

Accepted Answer

The answer of During its manufacture, a product is subjected...

Question 10

In any production process in which one or more workers are engaged in a variety of tasks, the total time spent in production varies as a function of the size of the workpool and the level of output of the various activities. In a large metropolitan department store, it is believed that the number of man-hours worked $( y )$ per day by the clerical staff depends on the number of pieces of mail processed per day $\left( x _ { 1 } \right)$ and the number of checks cashed per day $\left( x _ { 2 } \right)$. Data collected for $n = 20$ working days were used to fit the model:
$$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
A partial printout for the analysis follows:
Parameter Estimates
$\begin{array} { l r r r r r r } & & \text { PARAMETER } & \text { STANDARD } & \text { T FOR 0: } \ \text { VARIABLE } & \text { DF } & \begin{array} { r } \text { ESTIMATE } \end{array} & \begin{array} { r } \text { T FRROR } \end{array} & \text { PARAMETER = 0 } & \text { PROB } > | T | \ & & & & & \ \text { INTERCEPT } & 1 & 114.420972 & 18.6848744 & 6.124 & 0.0001 \ \text { X1 } & 1 & - 0.007102 & 0.00171375 & - 4.144 & 0.0007 \ \text { X2 } & 1 & 0.037290 & 0.02043937 & 1.824 & 0.0857 \end{array}$
Calculate a $95 \%$ confidence interval for $\beta _ { 1 }$.
A) $- .007 \pm .0036$
B) $- .007 \pm .0017$
C) $- .007 \pm .0007$
D) $- 4.144 \pm .0007$

Accepted Answer

The answer of In any production process in which one...

Question 11

In the first-order model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 } , \beta _ { 2 }$ represents the slope of the line relating $y$ to $x _ { 2 }$ when $\beta _ { 1 }$ and $\beta _ { 3 }$ are both held fixed.

Accepted Answer

The answer of In the first-order model \(E ( y...

Question 12

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:
Parameter Estimates
$\begin{array}{lrllll}
{\text { PARAMETER STANDARD }} & \text { T FOR 0: } \
\text { VARIABLE } & \text { DF } & \text { ESTIMATE } & \text { ERROR PARAMETER }=0 \text { PROB }>\text { |T| } =\
\text { INTERCEPT } 1 & -373.526392 & 1258.1243396-0.297 & 0.7676 \
\text { SPEED }\quad\quad\quad1 & 104.838940 & 22.36298195  4.688 & 0.0001 \
\text { CHIP } \quad\quad\quad\quad1 & 3.571850 & 3.89422935  0.917 & 0.3629
\end{array}$

Identify and interpret the estimate for the SPEED $\beta$-coefficient, $\hat { \beta } _ { 1 }$.
A) $\hat { \beta } _ { 1 } = 105$; For every 1-megahertz increase in SPEED, we estimate PRICE $( y )$ to increase $\$ 105$, holding CHIP fixed.
B) $\hat { \beta } _ { 1 } = 105$; For every $\$ 1$ increase in PRICE, we estimate SPEED to increase 105 megahertz, holding CHIP fixed.
C) $\hat { \beta } _ { 1 } = 3.57 ;$ For every 1 -megahertz increase in SPEED, we estimate PRICE to increase $\$ 3,57$, holding CHIP fixed.
D) $\hat { \beta } _ { 1 } = 3.57$; For every $\$ 1$ increase in PRICE, we estimate SPPED to increase by about 4 megahertz, holding CHIP fixed.

Accepted Answer

The answer of Retail price data for n = 60...

Question 13

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 )
$x _ { 2 } =$ 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 $$\quad $ $\quad $ $\quad $ $\quad $ $\quad $$\text { Parameter Estimates }$

$\quad $ $\quad $ $\quad $$\text { PARAMETER STANDARD\quad\quad\quad\quad T FOR 0: }$ 
$\text { VARIABLE DF ESTIMATE } \quad \text { ERROR PARAMETER }=0 \mathrm{PROB}>|\mathrm{T}|$ 
$\begin{array}{lrllll}
\text { INTERCEPT } & 1 & -373.526392 & 1258.1243396 & -0.297 & 0.7676 \
\text { SPEED } & 1 & 104.838940 & 22.36298195 & 4.688 & 0.0001 \
\text { CHIP } & 1 & 3.571850 & 3.89422935 & 0.917 & 0.3629
\end{array}$ 
Identify and interpret the estimate of $\beta _ { 2 }$.

Accepted Answer

The answer of Retail price data for n = 60...

Question 14

A first-order model does not contain any higher-order terms.

Accepted Answer

A first-order model includes only first-degree terms and does not include any higher-order terms.

Question 15

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

Accepted Answer

The answer of A first-order model may include terms for...

Question 16

The method of fitting first-order models is the same as that of fitting the simple straight-line model, i.e. the method of least squares.

Accepted Answer

The method of fitting first-order models involves using the method of least squares, which is also used for fitting simple straight-line models.

Question 17

The printout shows the results of a first-order regression analysis relating the sales price y of a product to the time in hours x1 and the cost of raw materials x2 needed to make the product. SUMMARY OUTPUT
$\begin{array}{ll}
\hline \text { Regression Statistics } & \
\hline \text { Multiple R } & 0.997578302 \
\text { R Square } & 0.995162468 \
\text { Adjusted R Square } & 0.990324936 \
\text { Standard Error } & 1.185250723 \
\text { Observations } & 5 \
\hline
\end{array}$

ANOVA
$\begin{array}{llllll} 
& d f & \text { SS } & \text { MS } & F & \text { Significance } F \
\hline \text { Regression } & 2 & 577.9903614 & 288.9952 & 205.717 & 0.004837532 \
\text { Residual } & 2 & 2.809638554 & 1.404819 & & \
\text { Total } & 4 & 580.8 & & & \
\hline
\end{array}$

$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & \text { tStat } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -26.48433735 & 3.674668773 & -7.20727 & 0.018713 & -42.29517198 & -10.67350271 \
\text { Time } & -2.168674699 & 4.11406532 & -0.52714 & 0.650732 & -19.8700814 & 15.532732 \
\text { Materials } & 8.142168675 & 1.094681583 & 7.437933 & 0.0176 & 3.432130693 & 12.85220666 \
\hline
\end{array}$
 a. What is the least squares prediction equation? b. Identify the SSE from the printout. c. Find the estimator of σ2 for the model.

Accepted Answer

The answer of The printout shows the results of a...

Question 18

A term that contains the value of a quantitative variable raised to the second power is called a higher -order term.

Accepted Answer

The answer of A term that contains the value of...

Question 19

Probabilistic models that include more than one dependent variable are called multiple regression models.

Accepted Answer

The answer of Probabilistic models that include more than one...

Question 20

A statistics professor gave three quizzes leading up to the first test in his class. The quiz grades and test grade for each of eight students are given in the table. $\begin{array}{ccccc}
\hline \text { Student }&\text {Test Grade  }&\text {Quiz } 1 &\text { Quiz } 2& \text { Quiz } 3\ 
\hline 1 & 75 & 8 & 9 & 5 \
2 & 89 & 10 & 7 & 6 \
3 & 73 & 9 & 8 & 7 \
4 & 91 & 8 & 7 & 10 \
5 & 64 & 9 & 6 & 6 \
6 & 78 & 8 & 7 & 6 \
7 & 83 & 10 & 8 & 7 \
8 & 71 & 9 & 4 & 6\
\hline  
\end{array}$
 The professor would like to use the data to find a first-order model that he might use to predict a student's grade on the first test using that student's grades on the first three quizzes. a. Identify the dependent and independent variables for the model. b. What is the least squares prediction equation? c. Find the SSE and the estimator of σ2 for the model. 2 Find and Interpret Sample Estimates for β Parameters

Accepted Answer

The answer of A statistics professor gave three quizzes leading...

Quiz 12: Multiple Regression and Model Building