Deck 17: Multiple Regression

Question 1

In multiple regression,the standard error of estimate is defined by ,where n is the sample size and k is the number of independent variables.

Answer 1

False

Question 2

When an additional explanatory variable is introduced into a multiple regression model,the coefficient of determination will never decrease.

Answer 2

True

Question 3

In order to test the significance of a multiple regression model involving 4 independent variables and 25 observations,the numerator and denominator degrees of freedom for the critical value of F are 3 and 21,respectively.

Answer 3

False

Question 4

In multiple regression analysis,the adjusted coefficient of determination is adjusted for the number of independent variables and the sample size.

Question 5

A multiple regression is called "multiple" because it has several explanatory variables.

Question 6

In reference to the equation ,the value 0.60 is the average change in y per unit change in x₂,regardless of the value of x₁.

Question 7

In reference to the equation ,the value −0.80 is the y-intercept.

Question 8

A multiple regression model involves 40 observations and 4 independent variables produces a total variation in y of 100,000 and SSR = 80,400.Then,the value of MSE is 560.

Question 9

In reference to the equation ,the value 0.12 is the average change in y per unit change in x₁,when x₂ is held constant.

Question 10

In regression analysis,the total variation in the dependent variable y,measured by ,can be decomposed into two parts: the explained variation,measured by SSR,and the unexplained variation,measured by SSE.

Question 11

When an additional explanatory variable is introduced into a multiple regression model,coefficient of determination adjusted for degrees of freedom can never decrease.

Question 12

In a multiple regression analysis involving 4 independent variables and 30 data points,the number of degrees of freedom associated with the sum of squares for error,SSE,is 25.

Question 13

In testing the significance of a multiple regression model with three independent variables,the null hypothesis is .

Question 14

In a multiple regression analysis involving 50 observations and 5 independent variables,the total variation in y is 475 and SSE = 71.25.Then,the coefficient of determination is 0.85.

Question 15

A multiple regression equation has a coefficient of determination of 0.81.Then,the percentage of the variation in y that is explained by the regression equation is 90%.

Question 16

In multiple regression analysis,when the response surface (the graphical depiction of the regression equation)hits every single point,the sum of squares for error SSE = 0,the standard error of estimate sε = 0,and the coefficient of determination R² = 1.

Question 17

Most statistical software print a second R² statistic,called the coefficient of determination adjusted for degrees of freedom,which has been adjusted to take into account the sample size and the number of independent variables.

Question 18

A multiple regression model is assessed to be good if the error sum of squares SSE and the standard error of estimate sε are both small,the coefficient of determination R² is close to 1,and the value of the test statistic F is large.

Question 19

A small value of F indicates that most of the variation in y is explained by the regression equation and that the model is useful.

Question 20

The coefficient of determination R² measures the proportion of variation in y that is explained by the explanatory variables included in the model.

Question 21

In multiple regression analysis,the ratio MSR/MSE yields the:

A)t-test statistic for testing each individual regression coefficient.
B)F-test statistic for testing the validity of the regression equation.
C)coefficient of determination.
D)adjusted coefficient of determination.

Question 22

The adjusted coefficient of determination is adjusted for the:

A)number of independent variables and the sample size.
B)number of dependent variables and the sample size.
C)coefficient of correlation and the significance level.
D)number of regression parameters including the y-intercept.

Question 23

The total variation in y in a regression model will never exceed the regression sum of squares (SSR).

Question 24

When an explanatory variable is dropped from a multiple regression model,the adjusted coefficient of determination can increase.

Question 25

A multiple regression model is assessed to be poor if the error sum of squares SSE and the standard error of estimate sε are both large,the coefficient of determination R² is close to 0,and the value of the test statistic F is large.

Question 26

When an explanatory variable is dropped from a multiple regression model,the coefficient of determination can increase.

Question 27

In order to test the validity of a multiple regression model involving 5 independent variables and 30 observations,the numerator and denominator degrees of freedom for the critical value of F are,respectively,

A)5 and 30
B)6 and 29
C)5 and 24
D)6 and 25

Question 28

A multiple regression model has the form .The coefficient b₁ is interpreted as the average change in y per unit change in x₁.

Question 29

In a multiple regression analysis involving 6 independent variables,the total variation in y is 900 and SSR = 600.What is the value of SSE?

A)300
B)1.50
C)0.67
D)None of these choices.

Question 30

In a multiple regression analysis,if the model provides a poor fit,this indicates that:

A)the coefficient of determination will be close to zero.
B)the standard error of estimate will be large.
C)the sum of squares for error will be large.
D)All of these choices are true.

Question 31

In calculating the standard error of the estimate, ,there are (n−k− 1)degrees of freedom,where n is the sample size and k is the number of independent variables in the model.

Question 32

A multiple regression model has the form: .As x₂ increases by one unit,holding x₁ constant,then the value of y will increase by:

A)7.25 units
B)6 units on average
C)2 units
D)None of these choices

Question 33

A multiple regression model has the form .As x₃ increases by one unit,with x₁ and x₂ held constant,the y on average is expected to:

A)increase by 1 unit.
B)increase by 12 units.
C)decrease by 4 units.
D)decrease by 16 units.

Question 34

A high value of the coefficient of determination significantly above 0 in multiple regression,accompanied by insignificant t-statistics on all parameter estimates,very often indicates a high correlation between independent variables in the model.

Question 35

Suppose a multiple regression analysis involving 25 data points has and SSE = 36.Then,the number of the independent variables must be:

A)3
B)4
C)5
D)6

Question 36

A multiple regression model involves 5 independent variables and a sample of 10 data points.If we want to test the validity of the model at the 5% significance level,the critical value is:

A)6.26
B)3.33
C)9.36
D)4.24

Question 37

From the coefficient of determination,we cannot detect the strength of the relationship between the dependent variable y and any individual independent variable.

Question 38

A multiple regression model involves 10 independent variables and 30 observations.If we want to test at the 5% significance level whether one of the coefficients is = 0 (vs.≠ 0)the critical value will be:

A)2.228
B)2.093
C)1.729
D)1.697

Question 39

In a multiple regression model,the mean of the probability distribution of the error variable ε is assumed to be:

A)k,where k is the number of independent variables included in the model.
B)1.0
C)0.0
D)None of these choices.

Question 40

In a multiple regression analysis involving k independent variables and n data points,the number of degrees of freedom associated with the sum of squares for error is:

A)k− 1
B)n−k
C)n− 1
D)n−k− 1

Question 41

In a multiple regression model,the value of the coefficient of determination has to fall between

A)−1 and +1.
B)0 and +1.
C)−1 and 0.
D)None of these choices.

Question 42

If all the points for a multiple regression model with two independent variables were right on the regression plane,then the coefficient of determination would equal:

A)0.
B)1.
C)2,since there are two independent variables.
D)None of these choices.

Question 43

A multiple regression analysis involving three independent variables and 25 data points results in a value of 0.769 for the unadjusted coefficient of determination.Then,the adjusted coefficient of determination is:

A)0.385
B)0.877
C)0.591
D)0.736

Question 44

For a multiple regression model,the following statistics are given: Total variation in y = 500,SSE = 80,and n = 25.Then,the coefficient of determination is:

A)0.84
B)0.16
C)0.3125
D)0.05

Question 45

To test the validity of a multiple regression model,we test the null hypothesis that the regression coefficients are all zero by applying the:

A)F-test
B)t-test
C)z-test
D)None of these choices.

Question 46

In a multiple regression analysis involving 40 observations and 5 independent variables,the following statistics are given: Total variation in y = 350 and SSE = 50.Then,the coefficient of determination is:

A)0.8408
B)0.8571
C)0.8469
D)0.8529

Question 47

For a multiple regression model the following statistics are given: Total variation in y = 250,SSE = 50,k = 4,and n = 20.Then,the coefficient of determination adjusted for the degrees of freedom is:

A)0.800
B)0.747
C)0.840
D)0.775

Question 48

A multiple regression model has the form .The coefficient b₁ is interpreted as the change in the average value of y per unit change in ________ holding ________ constant.

Question 49

In a multiple regression model,the following statistics are given: SSE = 100,R² = 0.995,k = 5,and n = 15.Then,the coefficient of determination adjusted for degrees of freedom is:

A)0.992
B)0.900
C)0.955
D)0.855

Question 50

In a multiple regression model,the error variable ε is assumed to have a mean of:

A)−1.0
B)0.0
C)1.0
D)None of these choices.

Question 51

For the multiple regression model: ,if x₂ were to increase by 5,holding x₁ and x₃ constant,the value of y will:

A)increase by 5.
B)increase by 75.
C)decrease on average by 5.
D)decrease on average by 75.

Question 52

Multiple regression has four requirements for the error variable.One is that the probability distribution of the error variable is ____________________.

Question 53

A multiple regression equation includes 5 independent variables,and the coefficient of determination is 0.81.The percentage of the variation in y that is explained by the regression equation is:

A)81%
B)90%
C)86%
D)about 16%

Question 54

The coefficient of determination ranges from:

A)1.0 to ∞.
B)0.0 to 1.0.
C)1.0 to k,where k is the number of independent variables in the model.
D)1.0 to n,where n is the number of observations in the dependent variable.

Question 55

A multiple regression model has:

A)only one independent variable.
B)only two independent variables.
C)more than one dependent variable.
D)more than one independent variable.

Question 56

For the following multiple regression model: ,a unit increase in x₁,holding x₂ and x₃ constant,results in:

A)a decrease of 3 units on average in the value of y.
B)an increase of 8 units in the value of y.
C)an increase of 3 units on average in the value of y.
D)None of these choices.

Question 57

In a multiple regression analysis,there are 20 data points and 4 independent variables,and the sum of the squared differences between observed and predicted values of y is 180.The standard error of estimate will be:

A)9.000
B)6.708
C)3.464
D)3.000

Question 58

In a multiple regression model,the probability distribution of the error variable ε is assumed to be:

A)normal.
B)non-normal.
C)positively skewed.
D)negatively skewed.

Question 59

For a multiple regression model,the total variation in y can be expressed as:

A)SSE − SSR.
B)SSR − SSE.
C)SSR + SSE.
D)SSR / SSE.

Question 60

In testing the validity of a multiple regression model in which there are four independent variables,the null hypothesis is:

A) .
B) .
C) .
D)
)

Question 61

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Interpret the coefficient b₁.

Question 62

Student's Final Grade
A statistics professor investigated some of the factors that affect an individual student's final grade in her course.She proposed the multiple regression model ,where y is the final grade (out of 100 points),x₁ is the number of lectures skipped,x₂ is the number of late assignments,and x₃ is the midterm exam score (out of 100).The professor recorded the data for 50 randomly selected students.The computer output is shown below. THE REGRESSION EQUATION IS ​ ​ ​ S = 13.74 R−Sq = 30.0% ​ ANALYSIS OF VARIANCE ​ ​
{Student's Final Grade Narrative} What is the coefficient of determination? What does this statistic tell you?

Question 63

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} What is the adjusted coefficient of determination in this situation? What does this statistic tell you?

Question 64

The computer output for the multiple regression model is shown below.However,because of a printer malfunction some of the results are not shown.These are indicated by the boldface letters a to i.Fill in the missing results (up to three decimal places). S = d
R−Sq = e
ANALYSIS OF VARIANCE

Question 65

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Is there sufficient evidence at the 5% significance level to infer that the number of points that the individual's blood pressure exceeded the recommended value and the age at death are negatively linearly related?

Question 66

We test an individual coefficient in a multiple regression model using a(n)_________ test.

Question 67

Consider the following statistics of a multiple regression model: Total variation in y = 1000,SSE = 300,n = 50,and k = 4.
a.Determine the standard error of estimate.
b.Determine the coefficient of determination.
c.Determine the F-statistic.

Question 68

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} What is the coefficient of determination? What does this statistic tell you?

Question 69

Some of the requirements for the error variable in a multiple regression model are that the probability distribution is ____________________ with a mean of ____________________.

Question 70

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Is there enough evidence at the 5% significance level to infer that the cholesterol level and the age at death are negatively linearly related?

Question 71

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Interpret the coefficient b₃.

Question 72

Consider the following statistics of a multiple regression model: n = 25,k = 5,b₁ = −6.31,and sε = 2.98.Can we conclude at the 1% significance level that x₁ and y are linearly related?

Question 73

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Is there enough evidence at the 1% significance level to infer that the average number of hours of exercise per week and the age at death are linearly related?

Question 74

The coefficient of determination ____________________ for degrees of freedom takes into account the sample size and the number of independent variables when assessing model fit.

Question 75

When there is more than one independent variable in a regression model,we refer to the graphical depiction of the equation as a(n)____________________ rather than as a straight line.

Question 76

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n)____________________ and the errors are ____________________.

Question 77

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Is there enough evidence at the 5% significance level to infer that the model is useful in predicting length of life?

Question 78

A(n)____________________ value of the F-test statistic indicates that the multiple regression model is valid.

Question 79

The total variation in y is equal to SSR + ____________________.

Question 80

Life Expectancy
An actuary wanted to develop a model to predict how long individuals will live.After consulting a number of physicians,she collected the age at death (y),the average number of hours of exercise per week (x₁),the cholesterol level (x₂),and the number of points that the individual's blood pressure exceeded the recommended value (x₃).A random sample of 40 individuals was selected.The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁− 0.021x₂− 0.061x₃ ​ S = 9.47 ​ R−Sq = 22.5% ANALYSIS OF VARIANCE ​ ​
{Life Expectancy Narrative} Interpret the coefficient b₂.