Question 1

Which residual plot would you examine to determine whether the assumption of constant error variance is satisfied for a model with two independent variables $x _ { 1 }$ and $x _ { 2 }$ ?
A) Plot the residuals against observed $y$ values.
B) Plot the residuals against the independent variable $x _ { 2 }$.
C) Plot the residuals against the independent variable $x _ { 1 }$.
D) Plot the residuals against predicted values, $\hat { y }$.

Accepted Answer

Plotting residuals against predicted values ($\hat{y}$) is the standard method to check for constant error variance (homoscedasticity) in a regression model, regardless of the number of independent variables.

Question 2

It is desired to build a regression model to predict $y =$ the sales price of a single family home, based on the $x _ { 1 } =$ size of the house and $x _ { 2 } =$ the neighborhood the home is located in. The goal is to compare the prices of homes that are located in two different neighborhoods. The following model is proposed:
$$\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 2 }$$
A regression model was fit and the following residual plot was observed.

Which of the following assumptions appears violated based on this plot?
A)The mean of the errors is zero
B)The variance of the errors is constant
C)The errors are normally distributed
D)The errors are independent

Accepted Answer

The answer of It is desired to build a regression...

Question 3

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?
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

Accepted Answer

The answer of A public health researcher wants to use...

Question 4

The staff of a test kitchen is attempting to determine the baking time, $ y $, of a roast, i.e., the time it takes the internal temperature of the roast to reach $ 165^{\circ} \mathrm{F} $, using two variables, the temperature setting of the oven, $ x_{1} $, and the weight of the roast, $ x_{2} $, in pounds. The data for 24 roasts are shown below.

$\text { Baking Times of Roasts }$

a. $\text { Fit a complete second-order model to the data. }$
b. Do the data provide sufficient evidence to indicate that the second-order terms contribute information for the prediction of $ y $ ? State the null and alternative hypothes

Accepted Answer

The answer of The staff of a test kitchen is...

Question 5

We expect all or almost all of the residuals to fall within 2 standard deviations of 0.

Accepted Answer

The answer of We expect all or almost all of...

Question 6

It is desired to build a regression model to predict $\mathrm { y } =$ the sales price of a single family home, based on the $x _ { 1 } =$ size of the house and $x _ { 2 } =$ the neighborhood the home is located in. The goal is to compare the prices of homes that are located in two different neighborhoods. The following complete 2nd-order model is proposed: $\mathrm { E } ( \mathrm { y } ) = \beta _ { 0 } + \beta _ { 1 } \mathrm { x } _ { 1 } + \beta _ { 2 } \mathrm { x } _ { 1 } ^ { 2 } + \beta _ { 3 } \mathrm { x } _ { 2 } + \beta _ { 4 } \mathrm { x } _ { 1 } \mathrm { x } _ { 2 } + \beta _ { 5 } \mathrm { x } _ { 1 } ^ { 2 } \mathrm { x } _ { 2 }$.
What hypothesis should be tested to determine if the quadratic terms are necessary to predict the sales price of a home?
A) $\mathrm { H } _ { 0 } : \beta _ { 2 } = \beta _ { 5 } = 0$
B) $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$
C) $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 3 } = \beta _ { 4 } = 0$
D) $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = \beta _ { 5 } = 0$

Accepted Answer

To determine if the quadratic terms ($x_1^2$ and $x_1^2x_2$) are necessary for predicting the sales price of a home, the hypothesis should test the significance of the coefficients associated with these terms. Therefore, $\mathrm { H } _ { 0 } : \beta _ { 2 } = \beta _ { 5 } = 0$ is the correct hypothesis to test the necessity of the quadratic terms in the model.

Question 7

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 increases because it involves multiple comparisons and model selection steps, each of which carries its own risk of error.

Question 8

The first-order model below was fit to a set of data. $$E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$$ Explain how to determine if the constant variance assumption is satisfied.

Accepted Answer

The answer of The first-order model below was fit to...

Question 9

The model $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ was fit to a set of data.
A partial printout for the analysis follows:

Interpret the value of the residual when $x _ { 1 } = 7,781$ and $x _ { 2 } = 644$.
A) The predicted $\hat { y }$ exceeds the observed value of $y$ by $8.468$.
B) Since the residual is not 0 , the model is not useful for predicting $y$.
C) The predicted $\hat { y }$ is $8.468$ less than the observed value of $y$.
D) Since the residual is negative, there is evidence of a negative linear relationship between $y$ and at least one of the two independent variables.

Accepted Answer

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

Question 10

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

Stepwise regression is a method of fitting regression models in which the choice of predictive variables is carried out by an automatic procedure, entering or removing predictors based on their statistical significance in predicting the dependent variable.

Question 11

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

Accepted Answer

The answer of There are four independent variables, \(x _...

Question 12

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

Accepted Answer

In linear regression models, the sum of the residuals (differences between observed and predicted values) is designed to be zero, as the best-fit line is calculated in a way that minimizes the sum of the squared residuals, leading to a balance where their sum equals zero.

Question 13

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.

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.

Accepted Answer

The answer of Suppose that the following model was fit...

Question 14

The stepwise regression procedure may not be used when the inclusion of one or more dummy
variables is under consideration.

Accepted Answer

Stepwise regression can be used when including dummy variables, as it is a method for selecting variables in a regression model, and dummy variables are a common way to include categorical data in such models.

Question 15

Residual analysis can be used to check for violations of the assumptions that the distribution of the
random error component is normally distributed with mean 0.

Accepted Answer

Residual analysis involves examining the residuals (the differences between observed and predicted values) to check if they are randomly distributed with a mean of 0, which is a key assumption for many statistical models, including the assumption of normally distributed errors.

Question 16

A regression residual is the difference between an observed y value and its corresponding
predicted value.

Accepted Answer

A regression residual is indeed the difference between an observed value of the dependent variable (y) and the value predicted by the regression model.

Question 17

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; a portion of the printout follows:

An outlier for the model is a clock with a residual that in absolute value. (Fill in the blank.)
A) exceeds 133
B) exceeds .893
C) is less than 266
D) exceeds 399

Accepted Answer

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

Question 18

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 $\alpha = 0.05$ ).
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.

Accepted Answer

The partial F-test comparing Model 1 and Model 2 resulted in a p-value of 0.0023, which is less than the significance level of 0.05. This indicates that the additional terms in Model 1 ($\beta_2x_1^2 + \beta_5x_1^2x_2$) significantly improve the model's fit, providing sufficient evidence of a quadratic relationship between sun-safety score and reading score.

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

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

Question 20

The stepwise regression model should not be used as the final model for predicting y.

Accepted Answer

Stepwise regression can introduce bias and overfitting due to its process of adding or removing predictors based solely on statistical criteria, without considering the underlying theory or causal relationships, leading to a model that may not generalize well to new data.

Quiz 12: Multiple Regression and Model Building

Which residual plot would you examine to determine whether the assumption of constant error variance is satisfied for a model with two independent variables $x _ { 1 }$ and $x _ { 2 }$ ?

We expect all or almost all of the residuals to fall within 2 standard deviations of 0.

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

The first-order model below was fit to a set of data. $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ Explain how to determine if the constant variance assumption is satisfied.

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

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

The stepwise regression procedure may not be used when the inclusion of one or more dummy variables is under consideration.

Residual analysis can be used to check for violations of the assumptions that the distribution of the random error component is normally distributed with mean 0.

A regression residual is the difference between an observed y value and its corresponding predicted value.

The stepwise regression model should not be used as the final model for predicting y.

Quiz 12: Multiple Regression and Model Building

Which residual plot would you examine to determine whether the assumption of constant error variance is satisfied for a model with two independent variables x1x _ { 1 }x1​ and x2x _ { 2 }x2​ ?

We expect all or almost all of the residuals to fall within 2 standard deviations of 0.

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

The first-order model below 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​ Explain how to determine if the constant variance assumption is satisfied.

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

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

The stepwise regression procedure may not be used when the inclusion of one or more dummy variables is under consideration.

Residual analysis can be used to check for violations of the assumptions that the distribution of the random error component is normally distributed with mean 0.

A regression residual is the difference between an observed y value and its corresponding predicted value.

The stepwise regression model should not be used as the final model for predicting y.

Which residual plot would you examine to determine whether the assumption of constant error variance is satisfied for a model with two independent variables $x _ { 1 }$ and $x _ { 2 }$ ?

The first-order model below was fit to a set of data. $E ( y ) = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ Explain how to determine if the constant variance assumption is satisfied.