Deck 9: Regression Analysis

Question 1

In regression modeling, the objective is to determines the values of model coefficients that minimize the sum of squared estimation errors, or error sum of squares (ESS).

Answer 1

True

Question 2

The term multicollinearity is used to describe the situation when the independent variables in a regression model are correlated among themselves.

Answer 2

True

Question 3

​In a model: Y_i = $\beta$ 0 + $\beta$ 1X_1i + $\varepsilon$ i , the terms $\beta$ 0 and $\beta$ 1 are referred to as sample statistics.

Answer 3

False

Question 4

Which of the following cannot be negative?

A) coefficient of determination
B) coefficient of correlation
C) coefficient of the independent variable, x, in the regression equation
D) y-intercept in the regression equation

Question 5

When using the Regression tool in Excel the independent variable is entered as the

A) X-range.
B) Y-range.
C) dependent-range.
D) independent-range.

Question 6

The problem of finding the optimal values of b₀ and b₁ is

A) a linear programming problem.
B) an unconstrained nonlinear optimization problem.
C) a goal programming problem.
D) a constrained nonlinear optimization problem.

Question 7

The forecasting model that makes use of the least squares method is called

A) regression
B) naive approach
C) moving average
D) exponential smoothing

Question 8

The actual value of a dependent variable will generally differ from the regression equation estimate due to

A) unaccounted for random variation.
B) the inability of the nonlinear Solver to find optimal values.
C) not building the regression model with enough data.
D) the model R² not equal to 1.

Question 9

​The R² statistic (also referred to as the coefficient of determination) ranges in value from 0 to 1 (0 $\le$ R² $\le$ 1) and indicates the proportion of the total variation in the dependent variable Y around its mean (average) that is accounted for by the independent variable(s) in the estimated regression function.

Question 10

​A simple linear regression model is of the form: Y_i = $\beta$ + $\beta$ 1X_1i + $\varepsilon$ i

Question 11

The value of adjusted ​R² can be negative.

Question 12

In regression analysis, the total variation is:

A) the sum of the squared deviations of each value of y from the mean of x
B) the sum of the explained variation and unexplained variation
C) the standard error of the forecast
D) equal to R²

Question 13

An analyst has identified 3 independent variables (X₁, X₂, X₃) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Why is the R² value for the X₃ model the same as the R² value for the X₁ and X₃ model, but the Adjusted R² values differ?

A) The standard error for X₁ is greater than the standard error for X₃.
B) X₁ does not reduce ESS enough to compensate for its addition to the model.
C) X₁ does not reduce TSS enough to compensate for its addition to the model.
D) X₁ and X₃ represent similar factors so multicollinearity exists.

Question 14

You want to conduct a hypothesis test for β₁. Based on the following regression output, what conclusion can you reach about β₁?

A) β₁ = 0, with P-value = 0.016353
B) β₁ ≠ 0, with P-value = 0.016353
C) β₁ = 0, with P-value = 0.000186
D) β₁ ≠ 0, with P-value = 0.000186

Question 15

The method of least squares finds estimates of parameter values that minimize:

A) TSS.
B) RSS.
C) ESS.
D) ESS + RSS.

Question 16

An analyst has identified 3 independent variables (X₁, X₂, X₃) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Which combination of variables provides the best regression results?

A) X₁
B) X₁, X₂ and X₃
C) X₁ and X₂
D) X₂ and X₃

Question 17

A residual is defined as the difference between the fitted value based on a model and a corresponding actual value.

Question 18

In regression terms what does "best fit" mean?

A) The estimated parameters, b₀ and b₁, are minimized.
B) The estimated parameters, b₀ and b₁, are linear.
C) The error terms are as small as possible.
D) The largest error term is as small as possible.

Question 19

​In regression analysis, we consider models of the form: Y = f(X₁, X₂, ..., X_k) + $\varepsilon$

Question 20

What is the formula for total sum of squares (TSS)

A)
B)
C)
D)

Question 21

What does regression analysis attempt to establish?

A) a mathematical relationship between a dependent variable, for which future values will be forecast, and one or more independent variables with known values
B) linearity in the relationship between independent variables
C) linearity in the relationship between a dependent variable and a set of independent variables
D) multicollinearity

Question 22

A persistent upward or downward movement of data is called

A) trend
B) seasonality
C) irregular variation
D) dampening signal

Question 23

The reason an analyst creates a regression model is

A) to determine the errors in the data collected.
B) to predict a dependent variable value given specific independent variable values.
C) to predict an independent variable value given specific dependent variable values.
D) to verify the errors are normally distributed.

Question 24

Error sum of squares (ESS) is computed as

A)
B)
C)
D)

Question 25

The regression line denotes the ____ between the dependent and independent variables.

A) unsystematic variation
B) systematic variation
C) random variation
D) average variation

Question 26

The regression residuals are computed as

A)
B)
C)
D)

Question 27

The term ε in the regression model represents

A) the slope of the regression model.
B) a random error term.
C) a correction for mistakes in measuring X.
D) a correction for the fact that we are taking a sample.

Question 28

Based on the following regression output, what is the equation of the regression line?

A)
B)
C)
D)

Question 29

The error sum of squares term is used as a criterion for determining b₀ and b₁ because

A) the sum of errors will always equal zero.
B) the term can be solved for exact values of b₀ and b₁.
C) both b₀ and b₁ can be easily calculated using the sum of squares term.
D) all of these.

Question 30

The regression function indicates the

A) average value the dependent variable assumes for a given value of the independent variable.
B) actual value the independent variable assumes for a given value of the dependent variable
C) average value the dependent variable assumes for a given value of the dependent variable
D) actual value the dependent variable assumes for a given value of the independent variable

Question 31

What goodness-of-fit measure is commonly used to evaluate a multiple regression function?

A) R²
B) adjusted R²
C) partial R²
D) total R²

Question 32

For a simple linear regression model, a 100(1 − α)% prediction interval for a new value of Y when X = X_h is computed as

A)
B)
C)
D)

Question 33

The estimated value of Y₁ is given by

A)
B)
C)
D)

Question 34

R² measures

A) the percentage of variability in the dependent variable, Y, explained by the model
B) the unexplained variability
C) the ratio of RSS/ESS
D) the model sophistication

Question 35

Based on the following regression output, what proportion of the total variation in Y is explained by X?

A) 0.917214
B) 0.841282
C) 0.821442
D) 9.385572

Question 36

Which of the following represents a regression model?

A)
B)
C) Y = f(X₁, X₂, ..., X_k)
D) Y = f(X₁, X₂, ..., X_k) + ε

Question 37

The standard prediction error is

A) always smaller than the standard error.
B) used to construct confidence intervals for predicted values.
C) measures the variability in the predicted values.
D) all of these.

Question 38

The R² statistic

A) varies between −1 and 1.
B) compares the regression sum of squares to the total sum of squares.
C) accounts for the number of parameters in the regression model.
D) is the ratio of the error sum of squares to the regression sum of squares.

Question 39

When using the Regression tool in Excel the dependent variable is entered as the

A) X-range.
B) Y-range.
C) dependent-range.
D) independent-range.

Question 40

The objective function in regression analysis is

A)
B)
C)
D)

Question 41

The total sum of squares (TSS) is best defined as

A) the sums of squares of the dependent variables.
B) the total variation of Y around its mean.
C) the sums of squares of the predicted values.
D) the variation of Y around its mean plus the variation of Y around the predicted values.

Question 42

In the equation Y = β₀ + β₁ X_1i + ε, β₁ is

A) the Y intercept
B) the slope of the regression line
C) the mean of the dependent data.
D) the X intercept

Question 43

Polynomial regression is used when

A) the independent variables are non-linear.
B) there is a non-linear relationship between the dependent and independent variables.
C) there is a non-linear relationship between the independent variables.
D) there is a curvilinear change in the dependent variables.

Question 44

R² is also referred to as

A) coefficient of determination.
B) correlation coefficient.
C) total sum of squares.
D) regression sum of squares.

Question 45

On average, the differences between the actual and predicted values of Y

A) are equal to b₀.
B) sum to an unknown value.
C) are distributed uniformly.
D) sum to zero.

Question 46

Based on the following regression output, what conclusion can you reach about β₀?

A) β₀ = 0, with P-value = 0.016353
B) β₀ ≠ 0, with P-value = 0.016353
C) β₀ = 0, with P-value = 0.000186
D) β₀ ≠ 0, with P-value = 0.000186

Question 47

How many independent variables are there in simple regression analysis?

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

Question 48

R² is calculated as

A) ESS/TSS
B) 1 − (RSS/TSS)
C) RSS/ESS
D) RSS/TSS

Question 49

The terms b₀ and b₁ are

A) estimated population parameters.
B) estimated intercept and slope values, respectively.
C) random variables.
D) all of these.

Question 50

Why do we create a scatter plot of the data in regression analysis?

A) To compute the error terms.
B) Because Excel calculates the function from the scatter plot.
C) To visually check for a relationship between X and Y.
D) To estimate predicted values.

Question 51

Based on the following regression output, what is the equation of the regression line?

A)
B)
C)
D)

Question 52

What is the correct range for R² values?

A) (−1 ≤ R² ≤ 0)
B) (−1 ≤ R² ≤ 1)
C) (0 ≤ R² ≤ 1)
D) (0 ≤ R² ≤ .5)

Question 53

The error term ε in a regression model represents

A) a random error in the data.
B) unsystematic variation in the dependent variable.
C) variation not explained by the independent variables.
D) all of these.

Question 54

The terms b₀ and b₁ are referred to as

A) population variables.
B) population parameters.
C) estimated population variables.
D) estimated population parameters.

Question 55

The β₁ term indicates

A) the average change in Y for a unit change in X.
B) the Y value for a given value of X.
C) the change in observed X for a given change in Y.
D) the Y value when X equals zero.

Question 56

A pattern resulting from random variation or unexplained causes is called

A) noise
B) trend
C) seasonality
D) time series

Question 57

Which of the following is an advantage of using the TREND() function versus the regression tool?

A) The TREND() function provides more statistical information.
B) The TREND() function handles multiple dependent variable data.
C) The TREND() function is dynamically updated when input to the function changes.
D) The TREND() function does not use a least squares regression line.

Question 58

The terms β₀ and β₁ are referred to as

A) sample statistics
B) random variables
C) population variables
D) population parameters

Question 59

What is a clear indicator of non-constant variance in a plot of regression model residuals?

A) A non-linear trend in the residual plot.
B) An intercept standard error larger that the estimated intercept coefficient.
C) A funnel shaped trend in the residual plot.
D) The standard errors from each independent variable differ.

Question 60

Residuals are assumed to be

A) dependent, uniformly distributed random variables.
B) independent, uniformly distributed random variables.
C) dependent, normally distributed random variables.
D) independent, normally distributed random variables.

Question 61

How many binary variables are required to encode a person's age group as being either young, middle-age or old? What are the variables and what are the meanings of their 0, 1 values?

Question 62

Exhibit 9.1
The following questions are based on the problem description and spreadsheet below.
A company has built a regression model to predict the number of labor hours (Y_i) required to process a batch of parts (X_i). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.1. Test the significance of the model and explain which values you used to reach your conclusions.

Question 63

Exhibit 9.3
The following questions are based on the problem description and spreadsheet below.
A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.3. Interpret the meaning of R square in cell B3 of the spreadsheet.

Question 64

Exhibit 9.1
The following questions are based on the problem description and spreadsheet below.
A company has built a regression model to predict the number of labor hours (Y_i) required to process a batch of parts (X_i). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.1. What is the estimated regression function for this problem? Explain what the terms in your equation mean.

Question 65

Estimation errors are often referred to as

A) mistakes.
B) constant errors.
C) residuals.
D) squared errors.

Question 66

Exhibit 9.6
The partial regression output below applies to the following questions.
Refer to Exhibit 9.6. What is the F-statistic value?

Question 67

The company would like to build a prediction interval on the pressure for a can with a temperature of 125 degrees. What formula should be entered in cells B17:F21 of the following spreadsheet to compute this prediction interval? Partial results of the Regression analysis of the data are provided below.

Question 68

Exhibit 9.3
The following questions are based on the problem description and spreadsheet below.
A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.3. What is the estimated regression function for this problem? Explain what the terms in your equation mean

Question 69

The standard error measures the

A) variability in the X values.
B) variability in the actual data around the fitted regression function.
C) variability in the independent variable around the fitted regression function.
D) variability in the dependent variable around the fitted regression function.

Question 70

Exhibit 9.1
The following questions are based on the problem description and spreadsheet below.
A company has built a regression model to predict the number of labor hours (Y_i) required to process a batch of parts (X_i). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.1. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet.

Question 71

The adjusted R² statistic

A) is equal to the value of unadjusted R²
B) adjusts R² for the degrees of freedom in the multiple regression model
C) accounts for the parameters in the multiple regression model
D) is always greater than R² unadjusted

Question 72

Exhibit 9.5
The following questions are based on the description and spreadsheet below.
An analyst has identified 3 independent variables (X₁, X₂,X₃) which might be used to predict Y. He has computed the regression equations using all of the variables and the results are summarized in the following table.
Refer to Exhibit 9.5. Predict the mean value based on (X₁, X₂, X₃) = (3, 32, 50). Use the best predictive model based on data from the table.

Question 73

Exhibit 9.1
The following questions are based on the problem description and spreadsheet below.
A company has built a regression model to predict the number of labor hours (Y_i) required to process a batch of parts (X_i). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.1. Provide a rough 95% confidence interval on the number of labor hours for a batch of 5 parts.

Question 74

Exhibit 9.2
The following questions are based on the problem description and spreadsheet below.
A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.2. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet.

Question 75

Exhibit 9.2
The following questions are based on the problem description and spreadsheet below.
A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.2. Interpret the meaning of R Square in cell B3 of the spreadsheet.

Question 76

Exhibit 9.6
The partial regression output below applies to the following questions.
Refer to Exhibit 9.6. What is the MS for Residual?

Question 77

Regression analysis is a modeling technique

A) that assumes all data is normally distributed.
B) for analyzing the relationship between dependent and independent variables.
C) for examining linear trend data only.
D) for capturing uncertainty in predicted values of Y.

Question 78

Exhibit 9.7
The partial regression output below applies to the following questions.
Refer to Exhibit 9.7. What is the SS for Residual and MS for Residual?

Question 79

Exhibit 9.3
The following questions are based on the problem description and spreadsheet below.
A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
Refer to Exhibit 9.3. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet.

Question 80

Exhibit 9.7
The partial regression output below applies to the following questions.
Refer to Exhibit 9.7. What is the SS for Total?