Question 1

The rejection of the null hypothesis in a global F-test means that the model is the best model for
providing reliable estimates and predictions.

Accepted Answer

A) True 
 B)False

Question 2

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$. Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$.

Accepted Answer

A)  16 
B)  10 
C)  13 
D)  1 
A)  16 
B)  10 
C)  13 
D)  1

Question 3

For any given model fit to a data set, the sum of the residuals is 0.

Accepted Answer

A) True 
 B)False

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 global $F$ statistic is used to test the null hypothesis, $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$. Describe this hypothesis in words.

Accepted Answer

A)  The model is not statistically useful for predicting Test4 score. 
B)  The first three test scores are poor predictors of Test 4 score. 
C)  The model is statistically useful for predicting Test4 score. 
D)  The first three test scores are reliable predictors of Test 4 score. 
A)  The model is not statistically useful for predicting Test4 score. 
B)  The first three test scores are poor predictors of Test 4 score. 
C)  The model is statistically useful for predicting Test4 score. 
D)  The first three test scores are reliable predictors of Test 4 score.

Question 5

It is desired to build a regression model to predict $\mathrm { y } =$ the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built?

Accepted Answer

A)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods. 
B)  $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$, where $x _ { 1 } - x _ { 3 }$ are qualitative variables that describe the four neighborhoods. 
C)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 } + \beta _ { 3 } \mathrm { x } _ { 3 } + \beta _ { 4 } \mathrm { x } _ { 4 }$, where $\mathrm { x } _ { 1 } - \mathrm { x } _ { 4 }$ are qualitative variables that describe the four neighborhoods. 
D)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 1 } ^ { 2 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods. 
A)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods. 
B)  $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 3 }$, where $x _ { 1 } - x _ { 3 }$ are qualitative variables that describe the four neighborhoods. 
C)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 } + \beta _ { 3 } \mathrm { x } _ { 3 } + \beta _ { 4 } \mathrm { x } _ { 4 }$, where $\mathrm { x } _ { 1 } - \mathrm { x } _ { 4 }$ are qualitative variables that describe the four neighborhoods. 
D)  $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 1 } ^ { 2 }$, where $\mathrm { x } _ { 1 }$ is a qualitative variable that describes the four neighborhoods.

Question 6

There are four independent variables, $x _ { 1 } , x _ { 2 } , x _ { 3 }$, and $x _ { 4 }$, that might be useful in predicting a response $y$. A total of $n = 40$ observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to useful. The computer fits all possible one-variable models of the form $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { i } , i = 1,2,3,4$. The information in the table is provided from the computer printout.

Which independent variable is declared the best one-variable predictor of $y$ ?

Accepted Answer

A)  $x _ { 1 }$ 
B)  $x _ { 2 }$ 
C)  $x _ { 3 }$ 
D)  $x _ { 4 }$ 
A)  $x _ { 1 }$ 
B)  $x _ { 2 }$ 
C)  $x _ { 3 }$ 
D)  $x _ { 4 }$

Question 7

Consider the data given in the table below. $\begin{array}{cc}
\hline \mathrm{X} & \mathrm{Y} \
\hline 1 & 4 \
2 & 6 \
2 & 5 \
3 & 7 \
4 & 7 \
4 & 6 \
5 & 4 \
5 & 5 \
6 & 3 \
\hline
\end{array}$

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.

Accepted Answer

a. A quadratic model seems to

Question 8

In the presence of multicollinearity, you should avoid making inferences about the parameters
based on the t-tests.

Accepted Answer

A) True 
 B)False

Question 9

A public health researcher wants to use regression to predict the sun safety knowledge of pre-school children. The researcher randomly sampled 35 preschoolers, assigned them to one of
Two groups, and then measured the following three variables: SUNSCORE: $\quad \mathrm { y } =$ Score on sun-safety comprehension test
READING: $\quad \mathrm { x } _ { 1 } =$ Reading comprehension score
GROUP: $\quad x _ { 2 } = 1$ if child received a Be Sun Safe demonstration, 0 if not
A regression model was fit and the following residual plot was observed.
 
 Which of the following assumptions appears violated based on this plot?

Accepted Answer

A) The mean of the errors is zero 
B) The errors are independent 
C) The errors are normally distributed 
D) The variance of the errors is constant 
A) The mean of the errors is zero 
B) The errors are independent 
C) The errors are normally distributed 
D) The variance of the errors is constant

Question 10

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 $x$.

Accepted Answer

A) True 
 B)False

Question 11

Suppose that the following model was fit to a set of data. $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$
The corresponding plot if residuals against predicted values $\hat { y }$ is shown. Interpret the plot.

Accepted Answer

A)  It appears that the variance of $\varepsilon$ is not constant. 
B)  It appears that the data contain an outlier. 
C)  The residuals appear to be randomly scattered so that no model modifications are necessary. 
D)  It appears that a quadratic model would be a better fit. 
A)  It appears that the variance of $\varepsilon$ is not constant. 
B)  It appears that the data contain an outlier. 
C)  The residuals appear to be randomly scattered so that no model modifications are necessary. 
D)  It appears that a quadratic model would be a better fit.

Question 12

Stepwise regression is used to determine which variables, from a large group of variables, are
useful in predicting the value of a dependent variable.

Accepted Answer

A) True 
 B)False

Question 13

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

Accepted Answer

A SECOND-ORDER (QUADRATIC)MODE

Question 14

A public health researcher wants to use regression to predict the sun safety knowledge of pre-school children. The researcher randomly sampled 35 preschoolers, assigned them to one of
Two groups, and then measured the following three variables: SUNSCORE: $\quad y =$ Score on sun-safety comprehension test
READING: $\quad \mathrm { x } _ { 1 } =$ Reading comprehension score
GROUP: $\quad x _ { 2 } = 1$ if child received a Be Sun Safe demonstration, 0 if not
The following two models were hypothesized:
$$\begin{array} { l } 
\text { Model 1: } E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 1 } ^ { 2 } + \beta _ { 3 } x _ { 2 } + \beta _ { 4 } x _ { 1 } x _ { 2 } + \beta _ { 5 } x _ { 1 } ^ { 2 } x _ { 2 } \
\text { Model 2: } E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 3 } x _ { 2 } + \beta _ { 4 } x _ { 1 } x _ { 2 }
\end{array}$$
A partial f-test was conducted to compare the two models and the resulting p-value was found to be $0.0023$. Fill in the blank. The results lead us to conclude that there is (at $\alpha = 0.05$ ).

Accepted Answer

A)  sufficient evidence of interaction between sun-safety score and reading score. 
B)  insufficient evidence of quadratic relationship between sun-safety score to reading score. 
C)  sufficient evidence of a quadratic relationship between sun-safety score to reading score. 
D)  sufficient evidence of a statistically useful model for sun-safety score. 
A)  sufficient evidence of interaction between sun-safety score and reading score. 
B)  insufficient evidence of quadratic relationship between sun-safety score to reading score. 
C)  sufficient evidence of a quadratic relationship between sun-safety score to reading score. 
D)  sufficient evidence of a statistically useful model for sun-safety score.

Question 15

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$.

Accepted Answer

A) True 
 B)False

Question 16

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:

Interpret the interval given in the printout.

Accepted Answer

A)  We are $95 \%$ confident that the average price of all hard drives with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$. 
B)  We are $95 \%$ confident that the price of a single hard drive falls between $\$ 3,943$ and $\$ 4,987$. 
C)  We are $95 \%$ confident that the average price of all hard drives falls between $\$ 3,943$ and $\$ 4,987 .$ 
D)  We are $95 \%$ confident that the price of a single hard drive with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$. 
A)  We are $95 \%$ confident that the average price of all hard drives with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$. 
B)  We are $95 \%$ confident that the price of a single hard drive falls between $\$ 3,943$ and $\$ 4,987$. 
C)  We are $95 \%$ confident that the average price of all hard drives falls between $\$ 3,943$ and $\$ 4,987 .$ 
D)  We are $95 \%$ confident that the price of a single hard drive with 33 megahertz speed and 386 CPU falls between $\$ 3,943$ and $\$ 4,987$.

Question 17

What relationship between x and y is suggested by the scattergram?

Accepted Answer

A)  a linear relationship with positive slope 
B)  a quadratic relationship with downward concavity 
C)  a quadratic relationship with upward concavity 
D)  a linear relationship with negative slope 
A)  a linear relationship with positive slope 
B)  a quadratic relationship with downward concavity 
C)  a quadratic relationship with upward concavity 
D)  a linear relationship with negative slope

Question 18

A college admissions officer proposes to use regression to model a studentʹs college GPA
at graduation in terms of the following two variables:
x1 = high school GPA
x2 = SAT score
The admissions officer believes the relationship between college GPA and high school
GPA is linear and the relationship between SAT score and college GPA is linear. She also
believes that the relationship between college GPA and high school GPA depends on the
studentʹs SAT score. She proposes the regression model: $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } x _ { 2 }$$ Explain how to determine if the relationship between college GPA and SAT score depends
on the high school GPA.

Accepted Answer

To determine if the relationsh

Question 19

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: $\quad y =$ Time spent (in hours) taking telephone orders during the day
ORDERS: $\quad x _ { 1 } =$ Number of telephone orders received during the day
WEEK: $\quad x _ { 2 } = 1$ weekday, 0 if Saturday or Sunday
Consider the complete 2 nd-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

To determine if the quadratic component

Question 20

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained.

Accepted Answer

It appears

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ . Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$ .

For any given model fit to a data set, the sum of the residuals is 0.

It is desired to build a regression model to predict $\mathrm { y } =$ the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built?

In the presence of multicollinearity, you should avoid making inferences about the parameters based on the t-tests.

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 $x$ .

Stepwise regression is used to determine which variables, from a large group of variables, are useful in predicting the value of a dependent variable.

Consider the data given in the table below. 1 7 2 6 2 5 3 5 3 4 4 4 4 3 4 2 5 4 5 5 6 6 Plot the data on a scattergram. Does a second-order model seem to be a good fit for the data? Explain.

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$ .

What relationship between x and y is suggested by the scattergram?

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained. $The model E ( y ) = \beta _ { 0 } + \beta _ { 1 } x was fit to a set of data, and the following plot of residuals against x values was obtained.$

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

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

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Categorical Data Analysis

Nonparametric Statistics Available Online

Filters

Exam 12: Multiple Regression and Model Building

The rejection of the null hypothesis in a global F-test means that the model is the best model for providing reliable estimates and predictions.

Consider the interaction model E(y)=7+3x1−4x2+5x1x2E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }E(y)=7+3x1​−4x2​+5x1​x2​ . Find the slope of the line relating E(y)E ( y )E(y) and x1x _ { 1 }x1​ when x2=2x _ { 2 } = 2x2​=2 .

For any given model fit to a data set, the sum of the residuals is 0.

It is desired to build a regression model to predict y=\mathrm { y } =y= the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built?

In the presence of multicollinearity, you should avoid making inferences about the parameters based on the t-tests.

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

Suppose that the following model was fit to a set of data. E(y)=β0+β1x1+β2x2E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }E(y)=β0​+β1​x1​+β2​x2​ The corresponding plot if residuals against predicted values y^\hat { y }y^​ is shown. Interpret the plot.

Stepwise regression is used to determine which variables, from a large group of variables, are useful in predicting the value of a dependent variable.

Consider the data given in the table below. 1 7 2 6 2 5 3 5 3 4 4 4 4 3 4 2 5 4 5 5 6 6 Plot the data on a scattergram. Does a second-order model seem to be a good fit for the data? Explain.

The independent variables x1x _ { 1 }x1​ and x2x _ { 2 }x2​ interact when the effect on E(y)E ( y )E(y) of a change in x1x _ { 1 }x1​ depends on x2x _ { 2 }x2​ .

What relationship between x and y is suggested by the scattergram?

The model E(y)=β0+β1xE ( y ) = \beta _ { 0 } + \beta _ { 1 } xE(y)=β0​+β1​x was fit to a set of data, and the following plot of residuals against xxx values was obtained.

Statistics, Data, and Statistical Thinking

Methods for Describing Sets of Data

Probability

Discrete Random Variables

Continuous Random Variables

Sampling Distributions

Inferences Based on a Single Sample: Estimation With Confidence Intervals

Inferences Based on a Single

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

Analysis of Variance: Comparing More Than Two Means

Simple Linear Regression

Categorical Data Analysis

Nonparametric Statistics Available Online

Filters

Consider the interaction model $E ( y ) = 7 + 3 x _ { 1 } - 4 x _ { 2 } + 5 x _ { 1 } x _ { 2 }$ . Find the slope of the line relating $E ( y )$ and $x _ { 1 }$ when $x _ { 2 } = 2$ .

It is desired to build a regression model to predict $\mathrm { y } =$ the sales price of a single family home, based on the neighborhood the home is located in. The goal is to compare the prices of homes that are located in four different neighborhoods. Which regression model should be built?

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 $x$ .

The independent variables $x _ { 1 }$ and $x _ { 2 }$ interact when the effect on $E ( y )$ of a change in $x _ { 1 }$ depends on $x _ { 2 }$ .

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x$ was fit to a set of data, and the following plot of residuals against $x$ values was obtained. $The model E ( y ) = \beta _ { 0 } + \beta _ { 1 } x was fit to a set of data, and the following plot of residuals against x values was obtained.$