Question 1

Consider the data given in the table below. $$\begin{array} { c c } 
\hline \mathrm { X } & \mathrm { Y } \
\hline 1 & 7 \
2 & 6 \
2 & 5 \
3 & 5 \
3 & 4 \
4 & 4 \
4 & 3 \
4 & 2 \
5 & 4 \
5 & 5 \
6 & 6 \
\hline
\end{array}$$ Plot the data on a scattergram. Does a second-order model seem to be a good fit for the data? Explain.

Accepted Answer

The answer of Consider the data given in the table...

Question 2

The table shows the profit y (in thousands of dollars) that a company made during a month when the price of its product was x dollars per unit.

$\begin{array}{cc}
\hline \text { Profit, } y & \text { Price, } x \
\hline 12 & 1.20 \
17 & 1.25 \
20 & 1.29 \
21 & 1.30 \
24 & 1.35 \
26 & 1.39 \
27 & 1.40 \
23 & 1.45 \
21 & 1.49 \
20 & 1.50 \
15 & 1.55 \
11 & 1.59 \
10 & 1.60 \
5 & 1.65 \
\hline
\end{array}$

a. Fit the model $y = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x 2 + \varepsilon$ to the data and give the least squares prediction equation.
b. Plot the fitted equation on a scattergram of the data.
c. Is there sufficient evidence of downward curvature in the relationship between profit and price? Use $\alpha = .05$.

Accepted Answer

The answer of The table shows the profit y (in...

Question 3

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?

Accepted Answer

The answer of The model \(E ( y ) =...

Question 4

The table below shows data for $n = 20$ observations.

$\begin{array}{ccc}
\hline \mathrm{y} & \mathrm{x} 1 & \mathrm{x} 2 \
\hline 18 & 3 & 8 \
23 & 5 & 10 \
15 & 2 & 7 \
31 & 6 & 12 \
24 & 4 & 9 \
28 & 5 & 11 \
17 & 2 & 7 \
19 & 3 & 8 \
30 & 7 & 10 \
28 & 5 & 8 \
14 & 3 & 6 \
32 & 7 & 11 \
17 & 2 & 8 \
24 & 5 & 10 \
26 & 6 & 11 \
27 & 6 & 11 \
21 & 3 & 6 \
31 & 7 & 13 \
19 & 2 & 8 \
25 & 5 & 10 \
\hline
\end{array}$

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.

Accepted Answer

The answer of The table below shows data for \(n...

Question 5

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.

$\begin{array}{ccccc}
\text { Student}& \text { TestGrade}& \text {Quiz 1 } & \text { Quiz 2 } & \text { Quiz 3 } \
\hline 1 & 75 & 8 & 9 & 5 \
2 & 89 & 10 & 7 & 6 \
3 & 73 & 9 & 8 & 7 \
4 & 91 & 8 & 7 & 10 \
5 & 64 & 9 & 6 & 6 \
6 & 78 & 8 & 7 & 6 \
7 & 83 & 10 & 8 & 7 \
8 & 71 & 9 & 4 & 6 \ 
\hline
\end{array}$
 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 threequizzes.
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 $\sigma ^ { 2 }$ for the model.

Accepted Answer

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

Question 6

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:
$\begin{array}{lrrrrr} 
& {\text { Analysis of Variance }} \
\text { SOURCE } & \text { DF } & \text { SS } & \text { MS } & \text { F VALUE } & \text { PROB > F } \
\text { MODEL } & 2 & 34593103.008 & 17296051.504 & 19.018 & 0.0001 \
\text { ERROR } & 57 & 51840202.926 & 909477.24431 & & \
\text { CTOTAL } & 59 & 86432305.933 & & &
\end{array}$

$\begin{array}{llll}
\text { ROOT MSE } & 953.66516 & \text { R-SQUARE } & 0.4002 \
\text { DEP MEAN } & 3197.96667 & \text { ADJ R-SQ } & 0.3792 \
\text { C.V. } & 29.82099 & &
\end{array}$

Test to determine if the model is adequate for predicting the price of a computer. Use $\alpha =$ $.01$.

Accepted Answer

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

Question 7

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

Accepted Answer

Adding new terms to a least squares regression model can help explain more of the variation in the data and therefore decrease the sum of squared errors.

Question 8

An elections officer wants to model voter turnout $( y )$ in a precinct as a function of the type of precinct.
Consider the model relating mean voter turnout, $E ( y )$, to precinct type:
$$\begin{array} { l l } 
E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } , \text { where } & x _ { 1 } = 1 \text { if urban, } 0 \text { if not } \
& x _ { 2 } = 1 \text { if suburban, } 0 \text { if not } \
& \text { (Base level = rural) }
\end{array}$$
Interpret the value of $\beta _ { 2 }$.
A) the difference between the mean voter turnout for suburban and rural precincts
B) the difference between the mean voter turnout for suburban and urban precincts
C) the mean voter turnout for suburban precincts
D) the rate of increase in voter turnout $( y )$ for suburban precincts, i.e., the slope of the $y - x _ { 2 }$ line

Accepted Answer

The answer of An elections officer wants to model voter...

Question 9

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state.

Write a model for mean voter turnout, E(y), as a function of type of election. 
A) $E ( y ) = \beta _ { 0 } + \beta 1 ^ { x }$, where $x = 1$ if national, 0 if state
B) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$, where $x =$ voter turnout
C) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$, where $x _ { 1 } = 1$ if national, 0 if not and $x _ { 2 } = 1$ if state, 0 if not
D) $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$, where $x =$ voter turnout

Accepted Answer

The correct model is E(y) = β0 + β1x, where x = 1 if national, 0 if state. This model captures the effect of the type of election on voter turnout, with β1 representing the difference in mean voter turnout between national and state elections.

Question 10

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ bu the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$.

Accepted Answer

In an interaction model, the effect of one predictor variable on the response variable depends on the level of another predictor variable, leading to different slopes for different values of the interacting variable.

Question 11

If when using the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$ we determine that interaction between and $x _ { 2 }$ is not significant, we can drop the $x _ { 1 } x _ { 2 }$ term from the model and use the simpler model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$.

Accepted Answer

If the interaction term $x_1 x_2$ is not statistically significant, it suggests that the interaction does not contribute to the model, allowing us to simplify the model by removing it.

Question 12

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.
A) The model is statistically useful for predicting Test4 score.
B) The model is not statistically useful for predicting Test4 score.
C) The first three test scores are reliable predictors of Test4 score.
D) The first three test scores are poor predictors of Test4 score.

Accepted Answer

The null hypothesis states that all the regression coefficients are equal to zero, which means that the model is not statistically useful for predicting Test4 score.

Question 13

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:
$\begin{array} { r r r r r r r r } 
\hline & & & \text { Actual } & \text { Predict } & & \text { Lower 95\% CL } & \text { Upper 95\% CL } \
\text { OBS } & \text { X1 } & \text { X2 } & \text { Value } & \text { Value } & \text { Residual } & \text { Predict } & \text { Predict } \
& & & & & & & \
1 & 7781 & 644 & 74.707 & 83.175 & - 8.468 & 47.224 & 119.126 \
\hline
\end{array}$

Interpret the 95% prediction interval for y shown on the printout.
A) We are 95% confident that the mean number of man-hours worked per day falls between 47.224 and 119.126 for all days in which 7,781 pieces of mail are processed and 644 checks are
Cashed.
B) We are 95% confident that the number of man-hours worked per day falls between 47.224 and 119.126.
C) We are 95% confident that between 47.224 and 119.126 man-hours will be worked during a single day in which 7,781 pieces of mail are processed and 644 checks are cashed.
D) We expect to predict number of man-hours worked per day to within an amount between 47.224 and 119.126 of the true value.

Accepted Answer

The 95% prediction interval provides a range within which we are 95% confident that the actual number of man-hours worked on a single day will fall, given the specific values of the predictors (7,781 pieces of mail and 644 checks cashed).

Question 14

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 the following data collected for 40 consecutive working days:
 TIME: $\quad\quad y =$ Time spent (in hours) taking telephone orders during the day
ORDERS: $\quad x _ { 1 } =$ Number of telephone orders received during the day
WEEK: $\quad\quad x _ { 2 } = 1$ weekday, 0 if Saturday or Sunday

Consider the complete 2nd-order model:
$$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } \left( x _ { 1 } \right) ^ { 2 } + \beta _ { 3 } x _ { 2 } + \beta _ { 4 } x _ { 1 } x _ { 2 } + \beta _ { 5 } \left( x _ { 1 } \right) ^ { 2 } x _ { 2 }$$ Explain how to conduct a test to determine if a quadratic relationship between total order time and the number of orders taken is necessary in the regression model above. Specify the null and alternative hypotheses that are to be tested.

Accepted Answer

The answer of Operations managers often use work sampling to...

Question 15

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.

$\begin{array}{ccccc}
\hline \text { Student Test Grade } & \text { Quiz } 1 & \text { Quiz } 2 & \text { Quiz } 3 \
\hline 1 & 75 & 8 & 9 & 5 \
2 & 89 & 10 & 7 & 6 \
3 & 73 & 9 & 8 & 7 \
4 & 91 & 8 & 7 & 10 \
5 & 64 & 9 & 6 & 6 \
6 & 78 & 8 & 7 & 6 \
7 & 83 & 10 & 8 & 7 \
8 & 71 & 9 & 4 & 6 \
\hline
\end{array}$
The professor fit a first-order model to the data that he intends to use to predict a student's grade on the first test using that student's grades on the first three quizzes.

Test the null hypothesis $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$ against the alternative hypothesis $H _ { \mathrm { a } }$ : at least one $\beta _ { i } \neq 0$. Use $\alpha = .05$. Interpret the result.

Accepted Answer

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

Question 16

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

Accepted Answer

In stepwise regression, the probability of making one or more Type I or Type II errors is not necessarily small. In fact, the use of stepwise regression can increase the likelihood of making Type I errors (false positives) or Type II errors (false negatives) due to the multiple comparisons involved. It is important to interpret the results of stepwise regression with caution and always verify the findings with additional analyses.

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 $x _ { 1 }$ and the cost of raw materials $x _ { 2 }$ 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
$\begin{array} { l l l l l l } 
& d f & \text { SS } & \text { MS } & F & \text { Significance F } \
\hline \text { Regression } & 2 & 577.9903614 & 288.9952 & 205.717 & 0.004837532 \
\text { Residual } & 2 & 2.809638554 & 1.404819 & & \
\text { Total } & 4 & 580.8 & & & \
\hline
\end{array}$

$\begin{array}{lllllll}
\hline & \text { Coefficients } & \text { Standard Error } & t \text { Stat } & P \text {-value } & \text { Lower 95\% } & \text { Upper 95\% } \
\hline \text { Intercept } & -26.48433735 & 3.674668773 & -7.20727 & 0.018713 & -42.29517198 & -10.67350271 \
\text { Time } & -2.168674699 & 4.11406532 & -0.52714 & 0.650732 & -19.8700814 & 15.532732 \
\text { Materials } & 8.142168675 & 1.094681583 & 7.437933 & 0.0176 & 3.432130693 & 12.85220666 \
\hline
\end{array}$

a. What is the least squares prediction equation?
b. Identify the SSE from the printout.
c. Find the estimator of $\sigma ^ { 2 }$ for the model.

Accepted Answer

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

Question 18

A collector of grandfather clocks believes that the price received for the clocks at an auction increases with the number of bidders, but at an increasing (rather than a constant) rate. Thus, the model proposed to best explain auction price (y, in dollars) by number of bidders (x) is the quadratic model $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x + \beta _ { 2 } x ^ { 2 }$$
This model was fit to data collected for a sample of 32 clocks sold at auction; the resulting estimate of $\beta _ { 1 }$ was $- .31$.
Interpret this estimate of $\beta _ { 1 }$.
A) We estimate the auction price will increase $\$ .31$ for each additional bidder at the auction.
B) $\beta 1$ is a shift parameter that has no practical interpretation.
C) We estimate the auction price will be $- \$ .31$ when there are no bidders at the auction.
D) We estimate the auction price will decrease $\$ .31$ for each additional bidder at the auction.

Accepted Answer

The answer of A collector of grandfather clocks believes that...

Question 19

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

Question 20

The concessions manager at a beachside park recorded the high temperature, the number of people at the park, and the number of bottles of water sold for each of 12 consecutiveSaturdays. The data are shown below.

$\begin{array}{ccc}
\hline \text { Bottles Sold Temperature }\left({ }^{\circ} \mathrm{F}\right) & \text { People } \
\hline 341 & 73 & 1625 \
425 & 79 & 2100 \
457 & 80 & 2125 \
485 & 80 & 2800 \
469 & 81 & 2550 \
395 & 82 & 1975 \
511 & 83 & 2675 \
549 & 83 & 2800 \
543 & 85 & 2850 \
537 & 88 & 2775 \
621 & 89 & 2800 \
897 & 91 & 3100 \
\hline
\end{array}$

a. Fit the model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$ to the data, letting $y$ represent the number of bottles of water sold, $x _ { 1 }$ the temperature, and $x _ { 2 }$ the number of people at the park.
b. Identify at least two indicators of multicollinearity in the model.
c. Comment on the usefulness of the model to predict the number of bottles of water sold on a Saturday when the high temperature is $103 ^ { \circ } \mathrm { F }$ and there are 3500 people at the park.

Accepted Answer

The answer of The concessions manager at a beachside park...

Quiz 12: Multiple Regression and Model Building

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?

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state. Write a model for mean voter turnout, E(y) , as a function of type of election.

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ bu the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$ .

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

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.

Quiz 12: Multiple Regression and Model Building

The sum of squared errors (SSE) of a least squares regression model decreases when new terms are added to the model.

An elections officer wants to model voter turnout (y) in a precinct as a function of type of election, national or state. Write a model for mean voter turnout, E(y) , as a function of type of election.

In an interaction model, the relationship between E(y)E ( y )E(y) and x1x _ { 1 }x1​ is linear for each fixed value of x2x _ { 2 }x2​ bu the slopes of the lines relating E(y)E ( y )E(y) and x1x _ { 1 }x1​ may be different for two different fixed values of x2x _ { 2 }x2​ .

In stepwise regression, the probability of making one or more Type I or Type II errors is quite small.

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.

In an interaction model, the relationship between $E ( y )$ and $x _ { 1 }$ is linear for each fixed value of $x _ { 2 }$ bu the slopes of the lines relating $E ( y )$ and $x _ { 1 }$ may be different for two different fixed values of $x _ { 2 }$ .