Deck 17: Multiple Regression

Question 1

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 1

False

Question 2

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 2

False

Question 3

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.

Answer 3

True

Question 4

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

Question 5

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 6

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

Question 7

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 8

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

Question 9

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 10

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

Question 11

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 s = 0, and the coefficient of determination R² = 1.

Question 12

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 13

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 14

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

Question 15

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

Question 16

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 17

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

Question 18

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 19

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 20

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

Question 21

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 22

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 23

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 24

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

Question 25

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 26

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 27

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 28

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 29

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

Question 30

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 31

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 32

In a multiple regression model, the mean of the probability distribution of the error variable 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 33

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

Question 34

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 35

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

Question 36

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

Question 37

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 38

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

Question 39

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 40

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. $\neq$ 0) the critical value will be:

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

Question 41

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 42

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 43

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

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

Question 44

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 45

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 46

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

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

Question 47

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 48

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 49

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 50

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

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

Question 51

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

Question 52

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 53

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 54

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 55

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 56

The coefficient of determination ranges from:

A) 1.0 to $\infty$ .
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 57

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 58

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 59

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 60

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 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
ANALYSIS OF VARIANCE


-{Life Expectancy Narrative} What is the adjusted coefficient of determination in this situation? What does this statistic tell you?

Question 62

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 63

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 ANALYSIS OF VARIANCE

{Student's Final Grade Narrative} What is the coefficient of determination? What does this statistic tell you?

Question 64

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
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 65

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 66

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

Question 67

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
ANALYSIS OF VARIANCE


-{Life Expectancy Narrative} Interpret the coefficient b₃.

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
ANALYSIS OF VARIANCE


-{Life Expectancy Narrative} What is the coefficient of determination? What does this statistic tell you?

Question 69

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

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
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

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

Question 72

Consider the following statistics of a multiple regression model: n = 25, k = 5, b₁ = -6.31, and s = 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
ANALYSIS OF VARIANCE


-{Life Expectancy Narrative} Interpret the coefficient b₂.

Question 74

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 75

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
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 76

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

Question 77

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).
ANALYSIS OF VARIANCE

Question 78

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
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 79

The validity of a multiple regression model is tested using a(n) _________ test.

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
ANALYSIS OF VARIANCE


-{Life Expectancy Narrative} Interpret the coefficient b₁.