Deck 11: Regression Analysis: Statistical Inference

Question 1

Which of the following is the relevant sampling distribution for regression coefficients?

A) normal distribution
B) t-distribution with n-1 degrees of freedom
C) t-distribution with n-1-k degrees of freedom
D) F-distribution with n-1-k degrees of freedom

Answer 1

t-distribution with n-1-k degrees of freedom

Question 2

Suppose you run a regression of a person's height on his/her right and left foot sizes, and you suspect that there may be multicollinearity between the foot sizes. What types of problems might you see if your suspicions are true?

A) "wrong" values for the coefficients for the left and right foot size
B) large p-values for the coefficients for the left and right foot size
C) small t-values for the coefficients for the left and right foot size
D) all of these choices

Answer 2

all of these choices

Question 3

The term autocorrelation refers to the observation that:

A) analyzed data refers to itself
B) sample is related too closely to the population
C) data are in a loop (values repeat themselves)
D) time series variables are usually related to their own past values

Answer 3

time series variables are usually related to their own past values

Question 4

In regression analysis, the ANOVA table analyzes:

A) the variation of the response variable Y
B) the variation of the explanatory variable X
C) the total variation of all variables
D) all of these choices

Question 5

The value k in the number of degrees of freedom, n-k-1, for the sampling distribution of the regression coefficients represents the:

A) sample size
B) population size
C) number of coefficients in the regression equation, including the constant
D) number of independent variables included in the equation

Question 6

In the standardized value , the symbol represents the:

A) mean of
B) variance of
C) standard error of
D) degrees of freedom of

Question 7

The appropriate hypothesis test for an ANOVA test is:

A)
B)
C)
D)

Question 8

The appropriate hypothesis test for a regression coefficient is:

A)
B)
C)
D) none of these choices

Question 9

There is evidence that the regression equation provides little explanatory power when the F-ratio:

A) is large
B) equals the regression coefficient
C) is small
D) is the constant

Question 10

The ANOVA table splits the total variation into two parts. They are the:

A) acceptable and unacceptable variation
B) adequate and inadequate variation
C) resolved and unresolved variation
D) explained and unexplained variation

Question 11

Which definition best describes parsimony?

A) explaining the most with the least
B) explaining the least with the most
C) being able to explain all of the change in the response variable
D) being able to predict the value of the response variable far into the future

Question 12

An error term represents the vertical distance from any point to the:

A) estimated regression line
B) population regression line
C) value of the Y's
D) mean value of the X's

Question 13

A scatterplot that exhibits a "fan" shape (the variation of Y increases as X increases) is an example of:

A) homoscedasticity
B) heteroscedasticity
C) autocorrelation
D) multicollinearity

Question 14

In regression analysis, multicollinearity refers to the:

A) response variables being highly correlated
B) explanatory variables being highly correlated
C) response variable(s) and the explanatory variable(s) being highly correlated with one another
D) response variables being highly correlated over time

Question 15

Which of the following is not one of the assumptions of regression?

A) There is a population regression line that joins the SDs of all possible distributions of results.
B) The response variable is normally distributed.
C) The standard deviation of the response variable increases as the explanatory variables increase.
D) The errors are probabilistically independent.

Question 16

Time series data often exhibits which of the following characteristics?

A) homoscedasticity
B) heteroscedasticity
C) autocorrelation
D) multicollinearity

Question 17

Another term for constant error variance is:

A) homoscedasticity
B) heteroscedasticity
C) autocorrelation
D) multicollinearity

Question 18

What is not one of the guidelines for including/excluding variables in a regression equation?

A) Look at the t-value and associated p-value.
B) Check whether the t-value is less than or greater than 1.0.
C) The variables are logically related to one another.
D) Use economic or physical theory to make the decision.
E) All of these choices are guidelines.

Question 19

Which statement is true regarding regression error, ε?

A) It is the same as a residual.
B) It can be calculated from the observed data.
C) It cannot be calculated from the observed data.
D) It is unbiased.

Question 20

The t-value for testing is calculated using which of the following equations?

A) n - k - 1
B)
C)
D)

Question 21

The objective typically used in the tree types of equation-building procedures is to:

A) find the equation with a small s_e
B) find the equation with a large R²
C) find the equation with a small s_e and a large R²
D) find the equation with the smallest F-ratio

Question 22

Which of the following would be considered a definition of an outlier?

A) an extreme value for one or more variables
B) a value whose residual is abnormally large in magnitude
C) values for individual explanatory variables that fall outside the general pattern of the other observations
D) all of these choices

Question 23

Determining which variables to include in regression analysis by estimating a series of regression equations by successively adding or deleting variables according to prescribed rules is referred to as:

A) elimination regression
B) forward regression
C) backward regression
D) stepwise regression

Question 24

Which approach can be used to test for autocorrelation?

A) regression coefficient
B) correlation coefficient
C) Durbin-Watson statistic
D) F-test or t-test

Question 25

When the error variance is nonconstant, it is common to see the variation increases as the explanatory variable increases (you will see a "fan shape" in the scatterplot). There are two ways you can deal with this phenomenon. These are:

A) the weighted least squares and a logarithmic transformation
B) the partial F and a logarithmic transformation
C) the weighted least squares and the partial F
D) stepwise regression and the partial F

Question 26

Suppose you forecast the values of all of the independent variables and insert them into a multiple regression equation and obtain a point prediction for the dependent variable. You could then use the standard error of the estimate to obtain an approximate:

A) confidence interval
B) prediction interval
C) hypothesis test
D) independence test

Question 27

In regression analysis, homoscedasticity refers to constant error variance.

Question 28

In regression analysis, extrapolation is performed when you:

A) attempt to predict beyond the limits of the sample
B) have to estimate some of the explanatory variable values
C) have to use a lag variable as an explanatory variable in the model
D) do not have observations for every period in the sample

Question 29

A point that "tilts" the regression line toward it, is referred to as a(n):

A) magnetic point
B) influential point
C) extreme point
D) explanatory point

Question 30

The assumptions of regression are: 1) there is a population regression line, 2) the dependent variable is normally distributed, 3) the standard deviation of the response variable remains constant as the explanatory variables increase, and 4) the errors are probabilistically independent.

Question 31

If you can determine that the outlier is not really a member of the relevant population, then it is appropriate and probably best to:

A) average it
B) reduce it
C) delete it
D) leave it

Question 32

When determining whether to include or exclude a variable in regression analysis, if the p-value associated with the variable's t-value is above some accepted significance value, such as 0.05, then the variable:

A) is a candidate for inclusion
B) is a candidate for exclusion
C) is redundant
D) does not fit the guidelines of parsimony

Question 33

Multiple regression represents an improvement over simple regression because it allows any number of response variables to be included in the analysis.

Question 34

In a simple linear regression model, testing whether the slope of the population regression line could be zero is the same as testing whether or not the linear relationship between the response variable Y and the explanatory variable X is significant.

Question 35

Residuals separated by one period that are autocorrelated indicate:

A) simple autocorrelation
B) redundant autocorrelation
C) time 1 autocorrelation
D) lag 1 autocorrelation

Question 36

A researcher can check whether the errors are normally distributed by using:

A) a t-test or an F-test
B) the Durbin-Watson statistic
C) a frequency distribution or the value of the regression coefficient
D) a histogram or a Q-Q plot

Question 37

Forward regression:

A) begins with all potential explanatory variables in the equation and deletes them one at a time until further deletion would do more harm than good
B) adds and deletes variables until an optimal equation is achieved
C) begins with no explanatory variables in the equation and successively adds one at a time until no remaining variables make a significant contribution
D) randomly selects the optimal number of explanatory variables to be used

Question 38

In time series data, errors are often not probabilistically independent.

Question 39

Many statistical packages have three types of equation-building procedures. They are:

A) forward, linear, and non-linear
B) forward, backward, and stepwise
C) simple, complex, and stepwise
D) inclusion, exclusion, and linear

Question 40

If exact multicollinearity exists, redundancy exists in the data.

Question 41

A forward procedure is a type of equation building procedure that begins with only one explanatory variable in the regression equation and successively adds one variable at a time until no remaining variables make a significant contribution.

Question 42

In multiple regression, the problem of multicollinearity affects the t-tests of the individual coefficients as well as the F-test in the analysis of variance for regression, since the F-test combines these t-tests into a single test.

Question 43

​In multiple regressions, if the F-ratio is large, the explained variation is large relative to the unexplained variation.

Question 44

In simple linear regression, if the error variable is normally distributed, the test statistic for testing is t-distributed with n - 2 degrees of freedom.

Question 45

When there is a group of explanatory variables that are in some sense logically related, all of them must be included in the regression equation.

Question 46

In a simple linear regression problem, if the standard error of estimate = 15 and n = 8, then the sum of squares for error, SSE, is 1,350.

Question 47

The residuals are observations of the error variable . Consequently, the minimized sum of squared deviations is called the sum of squared error, labeled SSE.

Question 48

Multicollinearity is a situation in which two or more of the explanatory variables are highly correlated with each other.

Question 49

In order to test the significance of a multiple regression model involving 4 explanatory variables and 40 observations, the numerator and denominator degrees of freedom for the critical value of F are 4 and 35, respectively.

Question 50

In multiple regression with k explanatory variables, the t-tests of the individual coefficients allows us to determine whether (for i = 1, 2, …., k), which tells us whether a linear relationship exists between and Y.

Question 51

In regression analysis, the unexplained part of the total variation in the response variable Y is referred to as the sum of squares due to regression, SSR.

Question 52

A multiple regression model involves 40 observations and 4 explanatory variables produces SST = 1000 and SSR = 804. The value of MSE is 5.6.

Question 53

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

Question 54

In testing the overall fit of a multiple regression model in which there are three explanatory variables, the null hypothesis is .

Question 55

In multiple regression, if there is multicollinearity between independent variables, the t-tests of the individual coefficients may indicate that some variables are not linearly related to the dependent variable, when in fact, they are.

Question 56

In a multiple regression analysis involving 4 explanatory variables and 40 data points, the degrees of freedom associated with the sum of squared errors, SSE, is 35.

Question 57

In multiple regressions, if the F-ratio is small, the explained variation is small relative to the unexplained variation.

Question 58

Suppose that one equation has 3 explanatory variables and an F-ratio of 49. Another equation has 5 explanatory variables and an F-ratio of 38. The first equation will always be considered a better model.

Question 59

The value of the sum of squares due to regression, SSR, can never be larger than the value of the sum of squares total, SST.

Question 60

A backward procedure is a type of equation building procedure that begins with all potential explanatory variables in the regression equation and deletes them two at a time until further deletion would reduce the percentage of variation explained to a value less than 0.50.

Question 61

A confidence interval constructed around a point prediction from a regression model is called a prediction interval, because the actual point being estimated is not a population parameter.

Question 62

A carpet company, which sells and installs carpet, believes that there should be a relationship between the number of carpet installations that they will have to perform in a given month and the number of building permits that have been issued within the county where they are located. Below you will find a regression model that compares the relationship between the number of monthly carpet installations (Y) and the number of building permits that have been issued in a given month (X). The data represents monthly values for the past 10 months.
(A) Estimate the regression model. How well does this model fit the given data?
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(B) Yes, there is a linear relationship between the number of carpet installations and the number of building permits issued at a = 0.10; The p-value = 0.0866 for the F-ratio. You can conclude that there is a significant linear relationship between these two variables.
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(C) The Durbin-Watson statistic for this data was 1.2183. Given this information what would you conclude about the data?
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(D) Given your answer in (C), would you recommend modifying the original regression model? If so, how would you modify it?

Question 63

Many companies manufacture products that are at least partially produced using chemicals (for example, paint). In many cases, the quality of the finished product is a function of the temperature and pressure at which the chemical reactions take place. Suppose that a particular manufacturer in Texas wants to model the quality (Y) of a product as a function of the temperature and the pressure at which it is produced. The table below contains data obtained from a designed experiment involving these variables. Note that the assigned quality score can range from a minimum of 0 to a maximum of 100 for each manufactured product.
(A) Estimate a multiple regression model that includes the two given explanatory variables. Assess this set of explanatory variables with an F-test, and report a p-value.
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(B) Identify and interpret the percentage of variance explained for the model in (A).
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(C) Identify and interpret the percentage of variance explained for the model in (B).
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(D) Which regression equation is the most appropriate one for modeling the quality of the given product? Bear in mind that a good statistical model is usually parsimonious.

Question 64

Below you will find a scatterplot of data gathered by an online retail company. The company has been able to obtain the annual salaries of their customers and the amount that each of these customers spent on the company's site last year. Based on the scatterplot below, would you conclude that these data meet all four assumptions of regression? Explain your answer.
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Question 65

The owner of a pizza restaurant chain would like to predict the sales of her recent specialty, a Mediterranean flatbread pizza. She has gathered data on monthly sales of the Mediterranean flatbread pizza at her restaurants. She has also gathered information related to the average price of flatbread pizzas, the monthly advertising expenditures, and the disposable income per household in the areas surrounding the restaurants. Below you will find output from the stepwise regression analysis. The p-value method was used with a cutoff of 0.05.
(A) Summarize the findings of the stepwise regression method using this cutoff value.
(B) When the cutoff value was increased to 0.10, the output below was the result. The table at top left represents the change when the disposable income variable is added to the model and the table at top right represents the average price variable being added. The regression model with both added variables is shown in the bottom table. Summarize the results for this model. (C) Which model would you recommend using? Why?

Question 66

A new online auction site specializes in selling automotive parts for classic cars. The founder of the company believes that the price received for a particular item increases with its age (i.e., the age of the car on which the item can be used in years) and with the number of bidders. The Excel multiple regression output is shown below.
(A) Estimate a multiple regression model for the data.
(B) Which of the variables in this model have regression coefficients that are statistically different from 0 at the 5% significance level?
(C) Given your findings in (B), which variables, if any, would you choose to remove from the model estimated in (A)? Explain your decision.

Question 67

The information below represents the relationship between the selling price (Y, in $1,000) of a home, the square footage of the home ( ), and the number of rooms in the home ( ). The data represents 60 homes sold in a particular area of East Lansing, Michigan and was analyzed using multiple linear regression and simple regression for each independent variable. The first two tables relate to the multiple regression analysis.
(A) Use the information related to the multiple regression model to determine whether each of the regression coefficients are statistically different from 0 at a 5% significance level. Summarize your findings.
(B) Test at the 5% significance level the relationship between Y and X in each of the simple linear regression models. How does this compare to your answer in (A)? Explain.
(C) Is there evidence of multicollinearity in this situation? Explain why or why not.

Question 68

One method of dealing with heteroscedasticity is to try a logarithmic transformation of the data.

Question 69

An Internet-based retail company that specializes in audio and visual equipment is interested in creating a model to determine the amount of money, in dollars, its customers will spend purchasing products from them in the coming year. In order to create a reliable model, this company has tracked a number of variables on its customers. Below you will find the Excel output related to several of these variables. This company has tried using the customer's annual salary for entire household , the number of children in the household , and if the customer purchased merchandise from them in the previous year ( in 2015).
(A) Estimate the regression model. How well does this model fit the data?
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(B) Is there a linear relationship between the explanatory variables and the dependent variable? Explain how you arrived at your answer at the 5% significance level.
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(C) Use the estimated regression model to predict the amount of money a customer will spend if their annual salary is $45,000, they have 1 child and they were a customer that purchased merchandise in the previous year (2015).
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(D) Find a 95% prediction interval for the point prediction calculated in (C). Use a t-multiple = 2.02.
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(E) Find a 95% confidence interval for the amount of money spent by all customers sharing the characteristics described in (C). Use a t-multiple = 2.02.
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(F) How do you explain the differences between the widths of the intervals in (D) and (E)?

Question 70

The owner of a large chain of health spas has selected eight of her smaller clubs for a test in which she varies the size of the newspaper ad , and the amount of the initiation fee discount to see how this might affect the number of prospective members who visit each club during the following week. The results are shown in the table below: The results of a multiple regression analysis are below.
(A) Determine the least-squares multiple regression equation.
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(B) Interpret the Y-intercept of the regression equation.
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(C) Interpret the partial regression coefficients.
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(D) What is the estimated number of new visitors to a club if the size of the ad is 6 column-inches and a $100 discount is offered?
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(E) Determine the approximate 95% prediction interval for the number of new visitors to a given club when the ad is 5 column-inches and the discount is $80.
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(F) What is the value for the percentage of variation explained, and exactly what does it indicate?
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(G) At the 0.05 level, is the overall regression equation in (A) significant?
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(H) Use the 0.05 level in concluding whether each partial regression coefficient differs significantly from zero.
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(I Interpret the results of the preceding tests in (H) and (I) in the context of the two explanatory variables described in the problem.
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(J) Construct a 95% confidence interval for each partial regression coefficient in the population regression equation.

Question 71

A company that makes baseball caps would like to predict the sales of its main product, standard little league caps. The company has gathered data on monthly sales of caps at all of its retail stores, along with information related to the average retail price, which varies by location. Below you will find regression output comparing these two variables.
(A) Estimate the regression model. How well does this model fit the given data?
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(B) Is there a linear relationship between X and Y at the 5% significance level? Explain how you arrived at your answer.
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(C) Use the estimated regression model to predict the number of caps that will be sold during the next month if the average selling price is $10.
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(D) Find a 95% prediction interval for the number of caps determined in (C). Use t- multiple = 2.
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(E) Find a 95% confidence interval for the average number of caps sold given an average selling price of $10. Use a t-multiple = 2.
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(F) How do you explain the differences between the widths of the intervals in (D) and (E)?

Question 72

In order to estimate with 90% confidence a particular value of Y for a given value of X in a simple linear regression problem, a random sample of 20 observations is taken. The appropriate t-value that would be used is 1.734.

Question 73

The manager of a commuter rail transportation system is asked by his governing board to predict the demand for rides in the large city served by the transportation network. The system manager has collected data on variables thought to be related to the number of weekly riders on the city's rail system. The table shown below contains these data. The variables "weekly riders" and "population" are measured in thousands, and the variables "price per ride", "income", and "parking rate" are measured in dollars.
(A) Estimate a multiple regression model using all of the available explanatory variables.
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(B) Conduct and interpret the result of an F- test on the given model. Employ a 5% level of significance in conducting this statistical hypothesis test.
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(C) Is there evidence of autocorrelated residuals in this model? Explain why or why not.

Question 74

Do you see any problems evident in the plot below of residuals versus fitted values from a multiple regression analysis? Explain your answer.

Question 75

A manufacturing firm wants to determine whether a relationship exists between the number of work-hours an employee misses per year (Y) and the employee's annual wages (X), to test the hypothesis that increased compensation induces better work attendance. The data provided in the table below are based on a random sample of 15 employees from this organization.
(A) Estimate a simple linear regression model using the sample data. How well does the estimated model fit the sample data?
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(B) Perform an F-test for the existence of a linear relationship between Y and X. Use a 5% level of significance.
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(C) Plot the fitted values versus residuals associated with the model. What does the plot indicate?
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(D) How do you explain the results you have found in (A) through (C)?
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(E) Suppose you learn that the 10^th employee in the sample has been fired for missing an excessive number of work-hours during the past year. In light of this information, how would you proceed to estimate the relationship between the number of work-hours an employee misses per year and the employee's annual wages, using the available information? If you decide to revise your estimate of this regression equation, repeat (A) and (B)

Question 76

One of the potential characteristics of an outlier is that the value of the dependent variable is much larger or smaller than predicted by the regression line.

Question 77

One method of diagnosing heteroscedasticity is to plot the residuals against the predicted values of Y, then look for a change in the spread of the plotted values.

Question 78

A local truck rental company wants to use regression to predict the yearly maintenance expense (Y), in dollars, for a truck using the number of miles driven during the year and the age of the truck in years at the beginning of the year. To examine the relationship, the company has gathered the data on 15 trucks and regression analysis has been conducted. The regression output is presented below.
(A) Estimate the regression model. How well does this model fit the given data?
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(B) Is there a linear relationship between the two explanatory variables and the dependent variable at the 5% significance level? Explain how you arrived at your answer.
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(C) Use the estimated regression model to predict the annual maintenance expense of a truck that is driven 14,000 miles per year and is 5 years old.
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(D) Find a 95% prediction interval for the maintenance expense determined in (C). Use a t-multiple = 2.
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(E) Find a 95% confidence interval for the maintenance expense for all trucks sharing the characteristics provided in (C). Use a t-multiple = 2.
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(F) How do you explain the differences between the widths of the intervals in (D) and (E)?

Question 79

The Durbin-Watson statistic can be used to test for autocorrelation.

Question 80

Which of the following is not one of the assumptions of regression?

A) There is a population regression line.
B) The response variable is not normally distributed.
C) The response variable is normally distributed.
D) The errors are probabilistically independent.