Deck 16: A: Simple Linear Regression and Correlation

Question 1

Statisticians have shown that sample y-intercept b₀ and sample slope coefficient b₁ are unbiased estimators of the population regression parameters $\beta$ ₀ and $\beta$ ₁,respectively.

Answer 1

True

Question 2

A regression analysis between weight (y in pounds)and height (x in inches)resulted in the following least squares line: $\hat { y } = 135 + 6 x$ .This implies that if the height is increased by 1 inch,the weight is expected to increase by an average of 6 pounds.

Answer 2

True

Question 3

Another name for the residual term in a regression equation is random error.

Answer 3

True

Question 4

The residual r_i is defined as the difference between the actual value y_i and the estimated value $\tilde { \gamma } _ { i }$ .

Question 5

A simple linear regression equation is given by $\hat { y } = 5.25 + 3.8 x$ .The point estimate of y when x = 4 is 20.45.

Question 6

An inverse relationship between an independent variable x and a dependent variably y means that as x increases,y decreases,and vice versa.

Question 7

A direct relationship between an independent variable x and a dependent variably y means that the variables x and y increase or decrease together.

Question 8

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: $\hat { y } = 77 + 8 x$ .This implies that if advertising is $600,then the predicted amount of sales (in dollars)is $125,000.

Question 9

The method of least squares requires that the sum of the squared deviations between actual y values in the scatter diagram and y values predicted by the regression line be minimized.

Question 10

If cov(x,y)= 7.5075 and $s _ { x } ^ { 2 } = 3.5$ ,then the sample slope coefficient is 2.145.

Question 11

To create a deterministic model,we start with a probabilistic model that approximates the relationship we want to model.

Question 12

A regression analysis between sales (in $)and advertising (in $)resulted in the following least squares line: $\hat { y } = 60 + 5 x$ .This implies that an increase of $1 in advertising is associated with an increase of $60 in sales.

Question 13

The regression line $\hat { y } = 3 + 2 x$ has been fitted to the data points (4,8), (2,5),and (1,2).The sum of the squared residuals will be:

A) 7
B) 15
C) 8
D) 22

Question 14

The residuals are observations of the error variable $\varepsilon$ .Consequently,the minimized sum of squared deviations is called the sum of squares for error,denoted SSE.

Question 15

The regression line $\hat { y } = 2 + 3 x$ has been fitted to the data points (4,11), (2,7),and (1,5).The sum of squares for error will be 10.0.

Question 16

The first-order linear model is sometimes called the simple linear regression model.

Question 17

A regression analysis between weight (y in pounds)and height (x in inches)resulted in the following least squares line: $\hat { y } = 120 + 5 x$ .This implies that if the height is increased by 1 inch,the weight,on average,is expected to:

A) increase by 1 pound.
B) decrease by 1 pound.
C) increase by 5 pounds.
D) increase by 24 pounds.

Question 18

The vertical spread of the data points about the regression line is measured by the y-intercept.

Question 19

Given the least squares regression line $\hat { y } = 5 - 2 x$ :

A) the relationship between x and y is positive.
B) the relationship between x and y is negative.
C) as x decreases,so does y.
D) None of these choices.

Question 20

If an estimated regression line has a y-intercept of 10 and a slope of 4,then when x = 2 the actual value of y is:

A) 18
B) 15
C) 14
D) unknown.

Question 21

The deviations between the actual data points and the fitted values from the model are called ____________________.

Question 22

In a simple linear regression model b₀ is the ____________________ of the straight line.

Question 23

SSE stands for ____________________ of squares for ____________________.

Question 24

In regression analysis,you predict the value of one variable on the basis of one or more other related variables.The variable being predicted is called the ____________________ variable,and the related variables used to make the prediction are called ____________________ variables.

Question 25

In a simple linear regression model b₁ is the ____________________ of the straight line.

Question 26

The objective of a regression model is to analyze the relationship between two variables,x and y,both of which must be based on ____________________ data.

Question 27

Suppose the slope of a simple linear regression line between hours studying and exam score is 5.That means as ____________________ increases by one,____________________ increases by 5.

Question 28

In regression analysis,the residuals represent the:

A) difference between the actual y values and their predicted values.
B) difference between the actual x values and their predicted values.
C) square root of the slope of the regression line.
D) change in y per unit change in x.

Question 29

In a simple linear regression problem,the following statistics are calculated from a sample of 10 observations: $$\sum ( x - \bar { x } ) ( y - \bar { y } ) = 2250 , s _ { x } = 10 , \sum x = 50 , \sum y = 75$$ .The least squares estimates of the slope and y-intercept are,respectively,

A) 1.5 and 0.5
B) 2.5 and 1.5
C) 1.5 and 2.5
D) 2.5 and -5.0

Question 30

In the first order linear regression model,the population parameters of the y-intercept and the slope are estimated,respectively,by:

A) b₀ and b₁
B) b₀ and $\beta$ ₁
C) $\beta$ ₀ and b₁
D) $\beta$ ₀ and $\beta$ ₁

Question 31

A straight line regression model with only one independent variable is called a(n)____________________-order linear model.

Question 32

The least squares method for determining the best fit minimizes:

A) total variation in the dependent variable
B) sum of squares for error
C) sum of squares for regression
D) All of these choices are true.

Question 33

In the simple linear regression model,the y-intercept represents the:

A) change in y per unit change in x.
B) change in x per unit change in y.
C) value of y when x = 0.
D) value of x when y = 0.

Question 34

In the least squares regression line $\hat { y } = 3 - 2 x$ ,the predicted value of y equals:

A) 1.0 when x = -1.0
B) 2.0 when x = 1.0
C) 2.0 when x = -1.0
D) 1.0 when x = 1.0

Question 35

In the simple linear regression model,the slope represents the:

A) value of y when x = 0.
B) average change in y per unit change in x.
C) value of x when y = 0.
D) average change in x per unit change in y.

Question 36

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: $\hat { y } = 75 + 6 x$ .This implies that if advertising is $800,then the predicted amount of sales (in dollars)is:

A) $4875
B) $123,000
C) $487,500
D) $12,300

Question 37

The residual is defined as the difference between:

A) the actual value of y and the estimated value of y
B) the actual value of x and the estimated value of x
C) the actual value of y and the estimated value of x
D) the actual value of x and the estimated value of y

Question 38

Which of the following techniques is used to predict the value of one variable on the basis of other variables?

A) Correlation analysis
B) Coefficient of correlation
C) Covariance
D) Regression analysis

Question 39

In the first-order linear regression model,the population parameters of the y-intercept and the slope are,respectively,

A) b₀ and b₁
B) b₀ and $\beta$ ₁
C) $\beta$ ₀ and b₁
D) $\beta$ ₀ and $\beta$ ₁

Question 40

A regression analysis between sales (in $1,000)and advertising (in $1,000)resulted in the following least squares line: $\hat { y } = 80 + 5 x$ .This implies that:

A) as advertising increases by $1,000,sales increases by $5,000.
B) as advertising increases by $1,000,sales increases by $80,000.
C) as advertising increases by $5,sales increases by $80.
D) None of these choices.

Question 41

Income and Education
A professor of economics wants to study the relationship between income (y in $1000s)and education (x in years).A random sample eight individuals is taken and the results are shown below.

{Income and Education Narrative} Determine the least squares regression line.

Question 42

Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below.

{Sunshine and Melanoma Narrative} What does the value of the slope of the regression line tell you?

Question 43

Sales and Experience
The general manager of a chain of department stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below.

{Sales and Experience Narrative} Determine the least squares regression line.

Question 44

A scatter diagram includes the following data points:
Two regression models are proposed: (1) ,and (2) .Using the least squares method,which of these regression models provides the better fit to the data? Why?

Question 45

You cannot interpret the ____________________ of the simple linear regression line unless the value of x = 0 lies within the range of where data was collected.

Question 46

Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below.

{Sunshine and Melanoma Narrative} Determine the least squares regression line.

Question 47

Speed vs Gas Mileage
An economist wanted to analyze the relationship between the speed of a car (x)and its gas mileage (y).As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured.These data are shown below.

{Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.

Question 48

Sales and Experience
The general manager of a chain of department stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below.

{Sales and Experience Narrative} Interpret the value of the slope of the regression line.

Question 49

Income and Education
A professor of economics wants to study the relationship between income (y in $1000s)and education (x in years).A random sample eight individuals is taken and the results are shown below.

{Income and Education Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

Question 50

The following 10 observations of variables x and y were collected.
Find the least squares regression line,and the estimated value of y when x = 3.

Question 51

Income and Education
A professor of economics wants to study the relationship between income (y in $1000s)and education (x in years).A random sample eight individuals is taken and the results are shown below.

{Income and Education Narrative} Estimate the income of an individual with 15 years of education.

Question 52

Sales and Experience
The general manager of a chain of department stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below.

{Sales and Experience Narrative} Estimate the monthly sales for a salesperson with 16 years of experience.

Question 53

Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below.

{Sunshine and Melanoma Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it.

Question 54

Sales and Experience
The general manager of a chain of department stores believes that experience is the most important factor in determining the level of success of a salesperson.To examine this belief she records last month's sales (in $1,000s)and the years of experience of 10 randomly selected salespeople.These data are listed below.

{Sales and Experience Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

Question 55

Speed vs Gas Mileage
An economist wanted to analyze the relationship between the speed of a car (x)and its gas mileage (y).As an experiment a car is operated at several different speeds and for each speed the gas mileage is measured.These data are shown below.

{Car Speed and Gas Mileage Narrative} Determine the least squares regression line.

Question 56

Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below.

{Sunshine and Melanoma Narrative} Estimate the number of skin cancer cases per 100,000 people who live in a state that gets 6 hours of sunshine on average.

Question 57

Income and Education
A professor of economics wants to study the relationship between income (y in $1000s)and education (x in years).A random sample eight individuals is taken and the results are shown below.

{Income and Education Narrative} Interpret the value of the slope of the regression line.

Question 58

Trivia Games & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Trivia Games & Education Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

Question 59

The method statisticians use to produce a straight line that minimizes the SSE is called the ____________________ method.

Question 60

Sunshine and Melanoma
A medical researcher wanted to examine the relationship between the amount of sunshine (x)in hours,and incidence of melanoma,a type of skin cancer (y).As an experiment he found the number of melanoma cases detected per 100,000 of population and the average daily sunshine in eight counties around the country.These data are shown below.

{Sunshine and Melanoma Narrative} Calculate the residual corresponding to the pair (x,y)= (8,15).

Question 61

Allman Brothers Concert
At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Allman Brothers Concert Narrative} Plot the least squares regression line on the scatter diagram.

Question 62

Cost of Textbooks
The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper.This implies that larger textbooks will cost more money.As an experiment to analyze the claim,a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks.These data are listed below.

{Cost of Textbooks Narrative} Estimate the selling price for a 650 pages book.

Question 63

Accidents and Rain
A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below.

{Accidents and Rain Narrative} What does the slope of the least squares regression line tell you?

Question 64

Accidents and Rain
A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below.

{Accidents and Rain Narrative} What other variables might be associated with accidents,besides or along with rain?

Question 65

Theatre Revenues
A financier whose specialty is investing in stage productions has observed that,in general,movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid performers in the play for ten recently staged plays.

{Theatre Revenues Narrative} Interpret the value of the slope of the regression line.

Question 66

Theatre Revenues
A financier whose specialty is investing in stage productions has observed that,in general,movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid performers in the play for ten recently staged plays.

{Theatre Revenues Narrative} Are the two highest paid performers worth all the money paid for them? Comment using the statistical analyses you have done.

Question 67

Allman Brothers Concert
At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Allman Brothers Concert Narrative} Determine the least squares regression line.

Question 68

Accidents and Rain
A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below.

{Accidents and Rain Narrative} Estimate the number of accidents in a day with 0.25 inches of rain.

Question 69

Trivia Games & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Trivia Games & Education Narrative} Interpret the value of the slope of the regression line.

Question 70

Cost of Textbooks
The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper.This implies that larger textbooks will cost more money.As an experiment to analyze the claim,a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks.These data are listed below.

{Cost of Textbooks Narrative} Determine the least squares regression line.

Question 71

Allman Brothers Concert
At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Allman Brothers Concert Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate to describe the relationship between the age and number of concerts attended by the respondents.

Question 72

Trivia Games & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Trivia Games & Education Narrative} Determine the least squares regression line.

Question 73

Trivia Games & Education
An ardent fan of television game shows has observed that,in general,the more educated the contestant,the less money he or she wins.To test her belief she gathers data about the last eight winners of her favorite game show.She records their winnings in dollars and the number of years of education.The results are as follows.

{Trivia Games & Education Narrative} Estimate the game winnings for a contestant with 15 years of education.

Question 74

Theatre Revenues
A financier whose specialty is investing in stage productions has observed that,in general,movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid performers in the play for ten recently staged plays.

{Theatre Revenues Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

Question 75

Cost of Textbooks
The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper.This implies that larger textbooks will cost more money.As an experiment to analyze the claim,a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks.These data are listed below.

{Cost of Textbooks Narrative} Draw a scatter diagram of the data.Comment on whether it appears that a linear model might be appropriate.

Question 76

Cost of Textbooks
The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper.This implies that larger textbooks will cost more money.As an experiment to analyze the claim,a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks.These data are listed below.

{Cost of Textbooks Narrative} Interpret the value of the slope of the regression line.

Question 77

Theatre Revenues
A financier whose specialty is investing in stage productions has observed that,in general,movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid performers in the play for ten recently staged plays.

{Theatre Revenues Narrative} Determine the least squares regression line.

Question 78

Allman Brothers Concert
At a recent Allman Brothers concert,a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year.The following data were collected:
An Excel output follows:
{Allman Brothers Concert Narrative} Interpret the value of the slope of the regression line.

Question 79

Theatre Revenues
A financier whose specialty is investing in stage productions has observed that,in general,movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known.To examine his belief he records the gross revenue and the payment (in $ millions)given to the two highest-paid performers in the play for ten recently staged plays.

{Theatre Revenues Narrative} Estimate the gross revenue of a play if the two highest paid performers received 6 million dollars.

Question 80

Accidents and Rain
A statistician investigating the relationship between the amount of rain (in inches)and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year.The results are shown below.

{Accidents and Rain Narrative} Find the least squares regression line.