Deck 15: Simple Linear Regression and Correlation

Question 1

True or False The value of the variation explained by the regression line can never be larger than 1.0.

Answer 1

False

Question 2

True or False The simple linear regression model assumes that regardless of the value for x,the standard deviation of the distribution of y values about the regression line is the same.

Answer 2

True

Question 3

True or False In a simple linear regression model,the residual is the horizontal distance from the regression line to an observed data point.

Answer 3

False

Question 4

True or False The coefficient of determination is a number that indicates both the direction and the strength of the linear relationship between the dependent and independent variable.

Question 5

True or False For a given data set of (x,y)values,an infinite number of possible regression equations can be fitted to the corresponding scatter diagram,and each equation will have a unique combination of values for the y-intercept b₀ and the slope b₁.However,only one equation will be the "best fit" as defined by the least-squares criterion.

Question 6

Another name for the residual term in a regression equation is:

A) random error.
B) heteroscedasticity.
C) homoscedasticity.
D) pooled variances.
E) residual analysis.

Question 7

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

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

Question 8

Simple linear regression requires that the scales of measurement be expressed in either:

A) nominal or ordinal.
B) ordinal or ratio.
C) interval or ratio.
D) nominal or ratio.
E) nominal or interval.

Question 9

True or False The simple linear regression model assumes that the y values are statistically independent of each other but the residuals are statistically dependent of each other.

Question 10

True or False The least-squares criterion requires that the sum of the squared deviations between the y values in the scatter diagram and the y values predicted by the equation be minimized.

Question 11

In the sample regression line $\hat { y } = b _ { 0 } + b _ { 1 } x$ the term b₀ is the y-intercept; this is the value of y where the line intersects the y-axis whenever x = 0.

Question 12

True or False When the predicted values of y and the actual values of y are the same,the standard error of estimate will be 0.0.

Question 13

True or False The simple linear regression model assumes that for any given value of x,the population of residuals will be normally distributed with a mean of zero and a standard deviation of 1.

Question 14

True or False An ANOVA test based on SST and SSR is equivalent to t-test for the coefficient of correlation and the slope.

Question 15

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: $\hat{y}$ = 75 +6x.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 16

Which of the following statements is true regarding the simple linear regression model $y _ { i } = \beta _ { 0 } + \beta _ { 1 } x _ { i } + \epsilon _ { 1 } ?$

A) is a value of the dependent variable (y) and x_i is a value of the independent variable (x).
B) $\beta$ ₀ is the y-intercept of the regression line.
C) $\beta$ ₁ is the slope of the regression line.
D) $\epsilon$ ₁ is a random error,or residual.
E) All of the above are true statements.

Question 17

Regardless of the value of x,the standard deviation of the distribution of y values about the regression line is the same.This assumption of equal standard deviations about the regression line is called:

A) random error.
B) heteroscedasticity.
C) homoscedasticity.
D) pooled variances.
E) residual analysis.

Question 18

A regression analysis between sales (in $1000)and advertising (in $)resulted in the following least squares line: $\hat{y}$ = 80,000 + 5x.This implies that an:

A) increase of $1 in advertising is expected to result in an increase of $5 in sales.
B) increase $5 in advertising is expected to result in an increase of $5,000 in sales.
C) increase of $1 in advertising is expected to result in an increase of $80,005 in sales.
D) increase of $1 in advertising is expected to result in an increase of $5,000 in sales.

Question 19

The residual is defined as the difference between the:

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

Question 20

True or False The coefficient of determination can be described in terms of the total variation in y versus the unexplained variation in y.

Question 21

If the standard error of estimate = 18 and n = 10, then the error sum of squares, SSE, is:

A) 2916.
B) 2592.
C) 1800.
D) 3240.

Question 22

The following values are listed as coefficients of correlation (r).The one that indicates an inverse relationship between the two variables x and y is:

A) 0.0.
B) -0.8.
C) 0.9.
D) 1.3.
E) -1.4.

Question 23

For a given value of x,the estimation interval for an individual y observation is called the:

A) confidence interval.
B) residual.
C) prediction interval.
D) least-squares interval.
E) standard error of estimate.

Question 24

Correlation analysis requires that the scales of measurement be expressed in either:

A) nominal or ordinal.
B) ordinal or ratio.
C) interval or ratio.
D) nominal or ratio.
E) nominal or interval.

Question 25

In simple linear regression,the coefficient of correlation r and the least squares estimate $b _ { 1 }$ of the population slope $\beta _ { 1 }$ :

A) must have the same numerical value.
B) must have opposite signs.
C) must have the same sign.
D) may have opposite signs or the same sign.

Question 26

Which of the following table values would be appropriate for a 95% confidence interval for the mean of y from a simple linear regression problem if the sample size is 7?

A) 1.895
B) 2.015
C) 2.365
D) 2.571
E) 1.960

Question 27

Given the least squares regression line $\hat{y}$ = -2.88 + 1.77x,and a coefficient of determination of 0.81,the coefficient of correlation is:

A) -0.88.
B) +0.88.
C) +0.90.
D) -0.90.

Question 28

The regression line = 3 + 2x has been fitted to the data points (4,8), (2,5), and (1,2). The residual sum of squares will be:

A) 10.
B) 15.
C) 13.
D) 22.

Question 29

The number of degrees of freedom associated with the standard error of estimate is

A) n-1 since only the slope is estimated from sample data.
B) n-1 since only the intercept is estimated from sample data.
C) n-1 since only the predicted value of y is estimated from sample data.
D) n-2 since the slope and the intercept are estimated from sample data.

Question 30

For the values of the coefficient of determination listed below,which one implies the greatest value of SSR (regression sum of squares)given that SST = 500?

A) 0.95
B) -1.00
C) 0.28
D) 0.00
E) -0.88

Question 31

In order to estimate with 95% confidence the expected value of y in a simple linear regression problem,a random sample of 10 observations is taken.Which of the following t-table values listed below would be used?

A) 2.228
B) 2.306
C) 1.860
D) 1.812

Question 32

In a regression problem the following pairs of (x,y)are given: (2,1),(2,-1),(2,0),(2,-2)and (2,2).That indicates that the:

A) coefficient of correlation is -1.
B) coefficient of correlation is 0.
C) coefficient of correlation is 1.
D) coefficient of determination is between -1 and 1.

Question 33

In publishing the results of some research work,the following values of the coefficient of determination were listed.Which one would appear to be incorrect?

A) 0.91
B) 0.06
C) 0.47
D) -0.64
E) 0.00

Question 34

All of the values of an independent variable equal the same number.Regressing a dependent variable on this independent variable will result in a coefficient of determination (r ² )of:

A) 0.0.
B) -1.0.
C) 2.3.
D) -2.3.
E) 1.0.

Question 35

The vertical spread of the data points about the regression line is measured by the:

A) regression coefficient.
B) standard error of estimate.
C) y-intercept.
D) homoscedasticity coefficient.
E) t-ratio.

Question 36

An indication of no linear relationship between two variables would be:

A) a coefficient of determination equal to 1.
B) a coefficient of determination equal to -1.
C) a coefficient of correlation of 0.
D) a coefficient of correlation equal to -1.
E) Both "A" and "B" are correct.

Question 37

The value for SSE equals zero.This means that the coefficient of determination (r ² )must equal:

A) 0.0.
B) -1.0.
C) 2.3.
D) -2.3.
E) 1.0.

Question 38

If all the points in a scatter diagram lie on the least squares regression line,then the coefficient of correlation:

A) must be 1.0.
B) must be -1.0.
C) must be either 1.0 or -1.0.
D) must be 0.

Question 39

Correlation analysis is used to determine the:

A) strength of the relationship between x and y.
B) least squares estimates of the regression parameters.
C) predicted value of y for a given value of x.
D) coefficient of determination.

Question 40

In a regression problem,if the coefficient of determination is 0.90,this means that:

A) 90% of the y values are positive.
B) 90% of the variation in y can be explained by the regression line
C) 90% of the x values are equal.
D) 90% of the variation in x can be explained by regression line.

Question 41

____________________ provides a "best-fit" mathematical equation for the values of two variables,x and y.

Question 42

Briefly describe each of the following:
a)Homoscedasticity
b)Residual

Question 43

Briefly describe each of the following:

A)An inverse relationship between variables __________________________________
B)A direct relationship between variables __________________________________
C)A linear relationship between variables __________________________________
D)A curvilinear relationship between variables __________________________________

Question 44

______________________________ measures the strength of the relationship between the dependent and independent variables.

Question 45

If the coefficient of correlation is either ____________________ or ____________________,then the regression line will actually include all of the data points and the line will be a perfect fit.

Question 46

The coefficient of determination can take on values between ____________________ and ____________________,inclusive.

Question 47

If the sum of squares due to regression (SSR)is 60,which of the following must be true?

A) The coefficient of correlation is 0.9.
B) The total sum of squares (SST)is at least 60.
C) The y-intercept is positive.
D) The slope,b,is positive.
E) The coefficient of determination is 0.81.

Question 48

Correlation analysis provides us with two important measures of the strength: (1)the coefficient of ____________________ and (2)the coefficient of ____________________.

Question 49

The ____________________ requires that the sum of the squared deviations between y values in the scatter diagram and y values predicted by the equation be minimized.

Question 50

The symbol for the independent variable is ____________________ and the symbol for the dependent variable is ____________________.

Question 51

The coefficient of correlation assumes values between ____________________ and ____________________,inclusive.

Question 52

The assumption of equal standard deviations about the regression line is called _________________________.

Question 53

A simple linear regression problem produced the following sum of squares: = 240, = 60, = 180
What percentage of the variation in y is explained by the regression line?

Question 54

If the coefficient of correlation is -0.90,what percentage of the variation in y is explained by the regression line?

Question 55

Consider the following data values of variables x and y
Use Excel or Minitab to construct a scatter diagram of the data points

Question 56

If the total variation in y values is 200,and the variation explained by regression line is 180,then the coefficient of determination is equal to ____________________.

Question 57

In a simple linear model,testing whether the slope of the population regression line is zero is the same as testing whether or not the population ____________________ equals zero.

Question 58

One way to examine whether two variables might be linearly related is to construct a ______________________________.

Question 59

If the coefficient of correlation r is positive,then the dependent variable and the independent variable x are said to be ____________________ related.However,if r is negative,then x and y are said to be ____________________ related.

Question 60

The variation explained by regression line is denoted by ____________________,while the variation not explained by regression line is denoted by ____________________.

Question 61

NARRBEGIN: Used cars
The following table shows the selling prices and mileages for 7 used cars of a certain model.

Using the least-squares regression line,predict the selling price of a car with 55,000 miles.

Question 62

NARRBEGIN: Used cars
The following table shows the selling prices and mileages for 7 used cars of a certain model.

Identify the dependent and independent variable.

Question 63

NARRBEGIN: Used cars
The following table shows the selling prices and mileages for 7 used cars of a certain model.

Find the least-squares regression line.

Question 64

NARRBEGIN: Teacher
A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course.

In testing the hypotheses vs. ,what is the value of the test statistic?

Question 65

NARRBEGIN: Teacher
A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course.

In testing the hypotheses vs. ,what is the conclusion at the 0.05 significance level?

Question 66

NARRBEGIN: GPA
The following table shows the grade point average (GPA)for 5 students along with their entrance exam scores for MBA programs (GMAT).Develop a model that would predict the GPA of a student based on their GMAT score.

Determine the standard error of estimate.

Question 67

NARRBEGIN: Number of years
Data was collected to describe the relationship between salary and number of years of working experience at a particular organization and is shown below in the following table.

Identify the dependent and independent variables.

Question 68

NARRBEGIN: GPA
The following table shows the grade point average (GPA)for 5 students along with their entrance exam scores for MBA programs (GMAT).Develop a model that would predict the GPA of a student based on their GMAT score.

Determine the least-squares regression line.

Question 69

NARRBEGIN: Teacher
A statistics teacher collected the following data to determine if the number of hours a student studied during the semester could be used to predict the final grade for the course.

In testing the hypotheses vs. ,what is the rejection region at the 0.05 significance level?

Question 70

NARRBEGIN: X and Y
Consider the following data values of variables x and y

Given that when a simple linear regression model applied to the data,it produced the following residuals:
0.062,-0.195,0.548,-0.682,0.089,-0.168,and 0.346.
Apply the Lilliefors test to the residuals at = 0.05.What is your conclusion?

Question 71

NARRBEGIN: X and Y
Consider the following data values of variables x and y

Given that the simple linear regression equation is ,use Minitab to create a normal probability plot of the residuals.

Question 72

NARRBEGIN: Used cars
The following table shows the selling prices and mileages for 7 used cars of a certain model.

Consider the following data values of variables x and y
What does the scatter diagram tell you about the relationship between x and y?

Question 73

What does the following scatter diagram tell you about the relationship between x and y?

Question 74

Construct a 95% prediction interval for an individual y value when x = 6.5.

Question 75

NARRBEGIN: Number of years
Data was collected to describe the relationship between salary and number of years of working experience at a particular organization and is shown below in the following table.

Find the least-squares regression line.

Question 76

NARRBEGIN: Used cars
The following table shows the selling prices and mileages for 7 used cars of a certain model.

Consider the following data values of variables x and y
Use Minitab or Excel to construct a scatter diagram of the data points and plot the least squares regression line on it.

Question 77

NARRBEGIN: Number of years
Data was collected to describe the relationship between salary and number of years of working experience at a particular organization and is shown below in the following table.

The divisor of the standard error of estimate in simple linear regression is:

Question 78

NARRBEGIN: Number of years
Data was collected to describe the relationship between salary and number of years of working experience at a particular organization and is shown below in the following table.

Using the least-squares regression line,predict the salary of an employee with 11 years of experience.

Question 79

Construct a 95% confidence interval for the mean GPA when GMAT = 6.5.

Question 80

A simple linear regression problem produced the following sum of squares: = 240, = 60, = 180
What percentage of the variation in y is not explained by the regression line?