Deck 14: Regression Analysis

Question 1

A residual is the difference between the observed and the predicted values of y.

Answer 1

True

Question 2

The coefficients of each explanatory variable in a multiple linear regression model have the same interpretation as the coefficient of the explanatory variable in a simple linear regression model.

Answer 2

False

Question 3

________ correlation can make two variables appear closely related when no casual relation exists.

Answer 3

Spurious

Question 4

A scatterplot graphically shows the relationship between two variables.

Question 5

The deterministic component of a linear regression model is due to the omission of relevant factors that influence the response variable.

Question 6

The deterministic component of the simple linear regression model is given by β₀ + β₁x.

Question 7

The correlation coefficient can only range between 0 and 1.

Question 8

Another name for an explanatory variable is the dependent variable.

Question 9

The sample correlation coefficient cannot equal zero.

Question 10

A residual is the difference between the predicted and observed values of y.

Question 11

Covariance can be used to determine if there is a linear relationship between two variables is positive or negative.

Question 12

If two linear regression models have the same number of explanatory variables, a model with an R² value of 0.45 is a better prediction model than a model with an R² value of 0.65.

Question 13

The actual value y may differ from the expected value E(y). Therefore, we add a ________ ________ term ε to develop a simple linear regression model.

Question 14

The covariance can be used to determine the strength of a linear relationship between two variables.

Question 15

Simple linear regression includes more than one explanatory variable.

Question 16

The correlation coefficient is unit-free.

Question 17

The covariance and correlation coefficient are measures that quantify the non-linear relationship between two variables.

Question 18

The correlation coefficient captures only a ________ relationship.

Question 19

The method of least squares picks the slope and intercept of the sample regression equation by minimizing SSE.

Question 20

The value 0.75 of a sample correlation coefficient indicates a stronger linear relationship than that of 0.60.

Question 21

A correlation coefficient r = −0.85 could indicate a ________.

A) very weak positive linear relationship
B) very strong negative linear relationship
C) very weak negative linear relationship
D) very strong positive linear relationship

Question 22

The following scatterplot indicates that the relationship between the two variables x and y is ________.

A) strong and positive
B) strong and negative
C) weak and negative
D) no relationship

Question 23

Consider the following simple linear regression model: y = β₀ + β₁+ ε. The response variable is ________.

A) y
B) x
C) ε
D) β₀

Question 24

What is the name of the variable that is used to predict another variable?

A) Response
B) Explanatory
C) Coefficient of determination
D) Standard error of the estimate

Question 25

With regression analysis, we explicitly assume that one variable, called the ________ variable, is influenced by other variable, called the explanatory variable.

Question 26

Which of the following statements is the least accurate concerning correlation analysis?

A) Correlation does not imply causation.
B) The correlation coefficient captures only a linear relationship.
C) The correlation coefficient may not be a reliable measure when outliers are present in one or both of the variables.
D) The correlation coefficient describes both the direction and strength of the relationship between two variables only if the two variables have the same units of measurement.

Question 27

A sample regression equation is given by = 155 - 34x₁ - 12x_2. If x₁ = 3 and x₂ = 2, the predicted value of y would be ________.

A) 29
B) 77
C) 233
D) 281

Question 28

Simple linear regression analysis differs from multiple regression analysis in that ________.

A) simple linear regression uses only one explanatory variable
B) the coefficient of correlation is meaningless in simple linear regression
C) goodness-of-fit measures cannot be calculated with simple linear regression
D) the coefficient of determination is always higher in simple linear regression

Question 29

A sample regression equation is given by = −100 + 0.5x. If x = 20, the predicted value of y is ________.

A) −80
B) −90
C) 110
D) 120

Question 30

The ________ regression model allows for multiple explanatory variables to predict the response variable.

Question 31

The variance of the rates of return is 0.25 for stock X and 0.01 for stock Y. The covariance between the returns of X and Y is −0.01. The correlation of the rates of return between X and Y is ________.

A) −0.25
B) −0.20
C) 0.20
D) 0.25

Question 32

We use ________ to derive the coefficient of determination.

Question 33

The numerical measure that gauges dispersion from the sample regression equation is the sample ________ of the residuals.

Question 34

The relationship between the response variable and the explanatory variables is ________ if the value of the response variable is uniquely determined by the explanatory variables.

Question 35

When testing whether the correlation coefficient differs from zero, the value of the test statistic is t₂₀ = 1.95 with a corresponding p-value of 0.0653. At the 5% significance level, can you conclude that the correlation coefficient differs from zero?

A) Yes, since the p-value exceeds 0.05.
B) Yes, since the test statistic value of 1.95 exceeds 0.05.
C) No, since the p-value exceeds 0.05.
D) No, since the test statistic value of 1.95 exceeds 0.05.

Question 36

The sample standard deviations for x and y are 10 and 15, respectively. The covariance between x and y is −120. The correlation coefficient between x and y is ________.

A) −0.8
B) −0.5
C) 0.5
D) 0.8

Question 37

The following scatterplot indicates that the relationship between the two variables x and y is ________.

A) weak and positive
B) strong and positive
C) strong and negative
D) no relationship

Question 38

When testing whether the correlation coefficient differs from zero, the value of the test statistic is t₂₀ = −2.95 with a corresponding p-value of 0.0061. At the 5% significance level, can you conclude that the correlation coefficient differs from zero?

A) Yes, since the p-value is less than 0.05.
B) Yes, since the absolute value of the test statistic exceeds 0.05.
C) No, since the p-value is less than 0.05.
D) No, since the absolute value of the test statistic exceeds 0.05.

Question 39

Which of the following identifies the range for a correlation coefficient?

A) Any value less than 1.
B) Any value greater than 0.
C) Any value between 0 and 1.
D) Any value between −1 and 1.

Question 40

If the ________ is substantially greater than zero and the number of explanatory variables is large compared with sample size, then the adjusted R² will differ substantially from R².

Question 41

Given the augmented Phillips model: y = β₀ + β₁x₁ + β₂x₂ + ε, where y = actual rate of inflation (%), x₁ = unemployment rate (%), and x₂ = anticipated inflation rate (%). The response variable or variables in this model is (are) the ________.

A) unemployment rate
B) actual inflation rate
C) anticipated inflation rate
D) unemployment rate and anticipated inflation rate

Question 42

The standard error of the estimate measures ________.

A) the standard deviation of the random error
B) the standard deviation of the response variable
C) the standard deviation of the explanatory variable
D) the standard deviation of the correlation coefficient

Question 43

Which of the following is not true of the standard error of the estimate?

A) It can take on negative values.
B) Theoretically, its value has no predefined upper limit.
C) It is a measure of the accuracy of the regression model.
D) It is based on the squared deviations between the actual and predicted values of the response variable.

Question 44

In the sample regression equation = b₀ + b₁x, What is ?

A) The y-intercept
B) The slope of the equation
C) The value of y when x = 0
D) The predicted value of y, given a specific x value

Question 45

In a simple linear regression model, if the points on a scatter diagram lie on a straight line with a negative slope, which of the following is the coefficient of determination?

A) −1
B) 0
C) +1
D) Infinity

Question 46

Given the augmented Phillips model: y = β₀ + β₁x₁ + β₂x₂ + ε, where y = actual rate of inflation (%), x₁ = unemployment rate (%), and x₂ = anticipated inflation rate (%). The explanatory variable or variables in this model is (are) the ________.

A) unemployment rate
B) actual inflation rate
C) anticipated inflation rate
D) unemployment rate and anticipated inflation rate

Question 47

Consider the sample regression equation = 10 - 5x, with an R² value of 0.65. Which of the following is the value of the sample correlation between y and ?

A) −0.81
B) −0.65
C) 0.81
D) 0.65

Question 48

In a simple linear regression model, if the plots on a scatter diagram lie on a straight line, which of the following is the standard error of the estimate?

A) −1
B) 0
C) +1
D) Infinity

Question 49

Suppose a sample regression equation is given by = 11 − 0.80x. What is the predicted value of y and residual when x is 2 and y is observed to be 9?

A) = 9.4, 0.4
B) = 9, -0.4
C) = 9, 0.4
D) - 9.4, -0.4

Question 50

Consider the following simple linear regression model: y = β₀ + β₁x + ε. The explanatory variable is ________.

A) y
B) x
C) ε
D) β₀

Question 51

Calculate the value of R² given the ANOVA portion of the following regression output:

A) 0.151
B) 0.515
C) 0.849
D) 1.000

Question 52

The capital asset pricing model is given by R - R_f = α + β(R_M - R_f) + ε, where R_M = expected return on the market, R_f = risk-free market return, and R = expected return on a stock or portfolio of interest. The response variable in this model is ________.

A) R
B) R - R_{f
C) RM
D) RM} - R_f

Question 53

Consider the following simple linear regression model: y = β₀ + β₁x + ε. β₀ and β₁ are ________.

A) the response variables
B) the random error terms
C) the unknown parameters
D) the explanatory variables

Question 54

The capital asset pricing model is given by R - R_f = α + β(R_M - R_f) + ε, where R_M = expected return on the market, R_f = risk-free market return, and R = expected return on a stock or portfolio of interest. The explanatory variable in this model is ________.

A) R
B) R - R_{f
C) RM
D) RM} - R_f

Question 55

Consider the following data: = 20, s_x = 2, = −5, s_y = 4, and b₁ = −0.8. Which of the following is the sample regression equation?

A) = −21 − 0.80x
B) = −21 + 0.80x
C) = 11 − 0.80x
D) = 11 + 0.80x

Question 56

Consider the following data: = 20, s_x = 2, = −5, s_y = 4, and b₁ = 0.40. Which of the following is the sample regression equation?

A) = −13 − 0.40x
B) = −13 + 0.40x
C) = 3 − 0.40x
D) = 3 + 0.40x

Question 57

Suppose a sample regression equation is given by = 3 + 0.40x. Suppose when x is 10 y is observed to be 8. What is the residual of the model prediction, and does the model under or overpredict the value of y?

A) = −1, the model overpredicts y
B) = 1, the model overpredicts y
C) = 1, the model underpredicts y
D) = −1, the model underpredicts y

Question 58

Consider the sample regression equation = 100 + 10x, with an R² value of 0.81. Which of the following is the value of the sample correlation between y and ?

A) −0.90
B) −0.81
C) 0.81
D) 0.90

Question 59

Consider the following simple linear regression model: y = β₀ + β₁x + ε. The random error term is ________.

A) y
B) x
C) ε
D) β₀

Question 60

The standard error of the estimate measures ________.

A) the variability of the explanatory variables
B) the variability of the values of the sample regression coefficients
C) the variability of the observed y-values around the predicted y-values
D) the variability of the predicted y-values around the mean of the observed y-values

Question 61

Consider the following sample regression equation = 150 - 20x, where y is the demand for Product A (in 1,000s) and x is the price of the product (in $). If the price of Product A is $5, then we expect demand to be ________.

A) 50
B) 500
C) 5,000
D) 50,000

Question 62

Consider the following sample regression equation = 150 - 20x, where y is the demand for Product A (in 1,000s) and x is the price of the product (in $). The slope coefficient indicates that if ________.

A) the price of Product A increases by $1, then we predict the demand to decrease 20
B) the price of Product A increases by $1, then we predict the demand to increase by 20
C) the price of Product A increases by $1, then we predict the demand to decrease by 20,000
D) the price of Product A increases by $1, then we predict the demand to increase by 20,000

Question 63

Unlike the coefficient of determination, the sample correlation coefficient in a simple linear regression ________.

A) can be greater than 1
B) measures the percentage of variation explained by the regression line
C) indicates whether the slope of the regression line is positive or negative
D) is a proportion

Question 64

A multiple regression model with two explanatory variables is estimated using 20 observations resulting in SSE = 550 and SST = 1000. The value of adjusted R² is the closest to ________.

A) 0.39
B) 0.45
C) 0.55
D) 0.61

Question 65

Consider the sample regression equation = 12 + 3x₁ - 5x₂ + 7x₃ - 2x₄. When x₁ increases by 1 unit and x₂ increases by 2 units, while x₃ and x₄ remain unchanged, what change would you expect in the predicted y?

A) Decrease by 2
B) Decrease by 4
C) Decrease by 7
D) No change in predicted y

Question 66

A statistics student is asked to estimate y = β₀ + β₁x + ε. She calculates the following values: = 440, = −568, , = 1,120, n = 11. Which of the following is the value of y if x equals 2?

A) 6.62
B) 11.78
C) 58.22
D) 63.38

Question 67

A statistics student is asked to estimate y = β₀ + β₁x + ε. She calculates the following values: = 440, = −568, , = 1,120, n = 11. Which of the following is the sample regression equation?

A) = 9.2 - 1.29x
B) = 9.2 + 1.29x
C) = 60.80 - 1.29x
D) = 60.80 + 1.29x

Question 68

A statistics student is asked to estimate y = β₀ + β₁x + ε. She calculates the following values: = 440, = −568, , = 1,120, n = 11. The value of the slope b₁ is ________.

A) −1.29
B) −0.51
C) 0.51
D) 1.29

Question 69

When two regression models applied on the same data set have the same response variable but a different number of explanatory variables, the model that would evidently provide the better fit is the one with a ________.

A) lower standard error of the estimate and a higher coefficient of determination
B) higher standard error of the estimate and a higher coefficient of determination
C) higher coefficient of determination and a lower adjusted coefficient of determination
D) lower standard error of the estimate and a higher adjusted coefficient of determination

Question 70

Over the past 30 years, the sample standard deviations of the rates of return for stock X and Stock Y were 0.20 and 0.12, respectively. The sample covariance between the returns of X and Y is 0.0096. When testing whether the correlation coefficient differs from zero, the value of the test statistic is t₂₈ = 2.31. At the 5% significance level, the critical value is t_0.025,28 = 2.048. The conclusion to the hypothesis test is ________.

A) to reject H₀; we can conclude that the correlation coefficient differs from zero
B) to reject H₀; we cannot conclude that the correlation coefficient differs from zero
C) do not reject H₀; we can conclude that the correlation coefficient differs from zero
D) do not reject H₀; we cannot conclude that the correlation coefficient differs from zero

Question 71

Over the past 30 years, the sample standard deviations of the rates of return for stock X and Stock Y were 0.20 and 0.12, respectively. The sample covariance between the returns of X and Y is 0.0096. To determine whether the correlation coefficient is significantly different from zero, the appropriate hypotheses are ________.

A) H₀: μ = 0, H_A: μ ≠ 0
B) H₀: ρ_xy = 0, H_A: ρ_xy ≠ 0
C) H₀: μ = 1, H_A: μ ≠ 1
D) H₀: ρ_xy = 1, H_A: ρ_xy ≠ 1

Question 72

A simple linear regression of the return of firm A, R_A, on the return of firm B, R_B, based on 18 observations, is = 2.2 + 0.4 R_B. If the coefficient of determination from this regression is 0.09, calculate the correlation between R_A and .

A) 0.03
B) 0.04
C) 0.09
D) 0.30

Question 73

Using the same data set, four models are estimated using the same response variable, however, the number of explanatory variables differs. Which of the following models provides the best fit?

A) Model 1
B) Model 2
C) Model 3
D) Model 4

Question 74

Consider the sample regression equation: = 12 + 2x₁ - 6x₂ + 6x₃ + 2x₄. When x₁ increases 1 unit and x₂ increases 2 units, while x₃ and x₄ remain unchanged, what change would you expect in the predicted y?

A) Decrease by 2
B) Decrease by 10
C) Increase by 2
D) No change in the predicted y

Question 75

A multiple regression model with four explanatory variables is estimated using 25 observations resulting in SSE = 660 and SST = 1,000. The value of adjusted R² is the closest to ________.

A) 0.21
B) 0.34
C) 0.66
D) 0.79

Question 76

A multiple regression model with two explanatory variables is estimated using 20 observations resulting in SSE = 550 and SST = 1000. Which of the following is the correct value of R²?

A) 0.10
B) 0.45
C) 0.55
D) 0.90

Question 77

Over the past 30 years, the sample standard deviations of the rates of return for stock X and Stock Y were 0.20 and 0.12, respectively. The sample covariance between the returns of X and Y is 0.0096. The correlation of the rates of return between X and Y is the closest to ________.

A) 0.20
B) 0.24
C) 0.36
D) 0.40

Question 78

A multiple regression model with four explanatory variables is estimated using 25 observations resulting in SSE = 660 and SST = 1,000. Which of the following is the correct value of R²?

A) 0.16
B) 0.34
C) 0.66
D) 0.84

Question 79

The coefficient of determination R² is ________.

A) sometimes negative
B) always lower than adjusted R^2
C) usually higher than adjusted R^2
D) always equal to adjusted R²

Question 80

The R² of a multiple regression of y on x₁ and x₂ measures the ________.

A) percentage of the variation in y that is explained by the variability of x₁
B) percentage of the variation in y that is explained by the variability of x₂
C) statistical significance of the coefficients in the sample regression equation
D) percentage of the variation in y that is explained by the sample regression equation