Deck 16: Simple Linear Regression and Correlat

Question 1

The value of the sum of squares for regression SSR can never be smaller than 1.

Answer 1

False

Question 2

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.

Answer 2

True

Question 3

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

Answer 3

True

Question 4

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

Question 5

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

Question 6

When the actual values y of a dependent variable and the corresponding predicted values are the same, the standard error of the estimate will be 1.0.

Question 7

If the coefficient of correlation is 1.0, then the coefficient of determination must be 1.0.

Question 8

The regression line 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 9

A regression analysis between sales (in $)and advertising (in $)resulted in the following least squares line: . This implies that an increase of $1 in advertising is associated with an increase of $60 in sales.

Question 10

The residual r_i is defined as the difference between the actual value y_i and the estimated value .

Question 11

The value of the sum of squares for regression SSR can never be smaller than 0.0.

Question 12

A simple linear regression equation is given by . The point estimate of y when x = 4 is 20.45.

Question 13

If cov( x , y )= 7.5075 and , then the sample slope coefficient is 2.145.

Question 14

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

Question 15

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

Question 16

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: . This implies that if advertising is $600, then the predicted amount of sales (in dollars)is $125,000.

Question 17

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 18

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

Question 19

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

Question 20

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

Question 21

If the coefficient of correlation is - 0.81, then the percentage of the variation in y that is explained by the regression line is 81%.

Question 22

A prediction interval is used when we want to predict a one-time occurrence for a particular value of y when the independent variable is a given x value.

Question 23

If the coefficient of determination is 0.95, this means that 95% of the variation in the independent variable x can be explained by the y variable.

Question 24

A store manager gives a pre-employment examination to new employees. The test is scored from 1 to 100. He has data on their sales at the end of one year measured in dollars. He wants to know if there is any linear relationship between pre-employment examination score and sales. An appropriate test to use is the t -test of the population correlation coefficient.

Question 25

If all the points in a scatter diagram lie on the least squares regression line, then the coefficient of correlation must be 1.0.

Question 26

If the coefficient of determination is 1.0, then the coefficient of correlation must be 1.0.

Question 27

If the error variable e is normally distributed, the test statistic for testing H ₀: b ₁ = 0 has a Student t -distribution with n - 2 degrees of freedom.

Question 28

If there is no linear relationship between two variables x and y , the coefficient of determination must be - 1.0.

Question 29

If the coefficient of determination is 0.95, this means that 95% of the y values were predicted correctly by the regression line.

Question 30

Correlation analysis is used to determine whether there is a linear relationship between an independent variable x and a dependent variable y .

Question 31

The probability distribution of the error variable e is normal, with mean E ( e )= 0, and standard deviation s _e =1.

Question 32

A zero correlation coefficient between a pair of random variables means that there is no linear relationship between the random variables.

Question 33

When the actual values y of a dependent variable and the corresponding predicted values are the same, the standard error of estimate s _e will be 0.0.

Question 34

A zero population correlation coefficient for x and y means that there is no type of relationship whatsoever between x and y.

Question 35

In a simple linear regression problem, the least squares line is , and the coefficient of determination is 0.81. The coefficient of correlation must be - 0.90.

Question 36

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

Question 37

The coefficient of determination is equal to the coefficient of correlation squared.

Question 38

If the value of the sum of squares for error SSE equals zero, then the coefficient of determination must equal zero.

Question 39

In a simple linear regression model, testing whether the slope b ₁ of the population regression line could be zero is the same as testing whether or not the population coefficient of correlation r equals zero.

Question 40

In simple linear regression, the denominator of the standard error of estimate s _e is .

Question 41

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 42

The regression line 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 43

A confidence interval estimate for the expected value of y will always be wider than the prediction interval for the same given value of x and the same confidence level.

Question 44

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 45

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 46

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 47

The graph of a confidence interval for the expected value of y is represented by two parallel lines, one on either side of the regression line.

Question 48

A regression analysis between weight ( y in pounds)and height ( x in inches)resulted in the following least squares line: . 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 49

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

A)b ₀ and b ₁
B)b ₀ and b ₁
C)b ₀ and b ₁
D)b ₀ and b ₁

Question 50

A confidence interval (as opposed to a prediction interval)is used to estimate the long-run average value of y .

Question 51

A regression analysis between sales (in $1000)and advertising (in $100)resulted in the following least squares line: . 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 52

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 53

A regression analysis between sales (in $1,000)and advertising (in $1,000)resulted in the following least squares line: . 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 54

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 55

There is more error in estimating a mean value of y as opposed to predicting an individual value of y .

Question 56

The confidence interval estimate of the expected value of y will be narrower than the prediction interval for the same given value of x and confidence level. This is because there is less error in estimating a mean value as opposed to predicting an individual value.

Question 57

Given the least squares regression line :

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 58

The prediction interval for a particular value of y is always wider than the confidence interval for mean value of y , given the same data set, x value, and confidence level.

Question 59

The point where confidence intervals and prediction intervals do best is .

Question 60

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

A)b ₀ and b ₁
B)b ₀ and b ₁
C)b ₀ and b ₁
D)b ₀ and b ₁

Question 61

In regression analysis, if the coefficient of determination is 1.0, then:

A)the sum of squares for error must be 1.0
B)the sum of squares for regression must be 1.0
C)the sum of squares for error must be 0.0
D)the sum of squares for regression must be 0.0

Question 62

When all the actual values of y are equal to their predicted values, the standard error of estimate will be:

A)1.0
B)- 1.0
C)0.0
D)None of these choices.

Question 63

Given that and n = 6, the standard error of estimate is:

A)3,749.00
B)937.25
C)30.61
D)None of these choices.

Question 64

In simple linear regression, most often we perform a two-tail test of the population slope b ₁ to determine whether there is sufficient evidence to infer that a linear relationship exists. The null hypothesis is stated as:

A)H ₀: b ₁ = 0
B)H ₀: b ₁ = b ₁
C)H ₀: b ₁ ¹ 0
D)None of these choices.

Question 65

Testing whether the slope of the population regression line could be zero is equivalent to testing whether the:

A)sample coefficient of correlation could be zero
B)standard error of estimate could be zero
C)population coefficient of correlation could be zero
D)sum of squares for error could be zero

Question 66

If the coefficient of correlation is - 0.60, then the coefficient of determination is:

A)- 0.60
B)- 0.36
C)0.36
D)0.77

Question 67

In the least squares regression line , 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 68

The coefficient of correlation is used to determine:

A)the strength and direction of the linear relationship between x and y .
B)the least squares estimates of the regression parameters.
C)the predicted value of y for a given value of x .
D)All of these choices.

Question 69

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

A)1.0
B)- 1.0
C)either 1.0 or - 1.0
D)0.0

Question 70

A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. The standard error of estimate was:

A)2.11
B)1.56
C)2.44
D)None of these choices.

Question 71

If the coefficient of determination is 0.975, then which of the following is true regarding the slope of the regression line?

A)All we can tell is that it must be positive.
B)It must be 0.975.
C)It must be 0.987.
D)Cannot tell the sign or the value.

Question 72

Given the least squares regression line , and a coefficient of determination of 0.81, the coefficient of correlation is:

A)- 0.66
B)0.81
C)- 0.90
D)0.90

Question 73

In a simple linear regression problem, the following statistics are calculated from a sample of 10 observations: . 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 74

In a simple linear regression problem, the following sum of squares are produced: , , and . The percentage of the variation in y that is explained by the variation in x is:

A)25%
B)75%
C)33%
D)50%

Question 75

The symbol for the population coefficient of correlation is:

A)r
B)r
C)r ²
D)r ²

Question 76

The symbol for the sample coefficient of correlation is:

A)r
B)r
C)r ²
D)r ²

Question 77

If the coefficient of correlation between x and y is close to 1.0, this indicates that:

A)y causes x to happen.
B)x causes y to happen.
C)both a and b.
D)there may or may not be a causal relationship between x and y.

Question 78

Given that the sum of squares for error is 60 and the sum of squares for regression is 140, then the coefficient of determination is:

A)0.429
B)0.300
C)0.700
D)None of these choices.

Question 79

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 80

If the coefficient of correlation is - 0.80, then the percentage of the variation in y that is explained by the variation in x is:

A)80%
B)64%
C)89%
D)None of these choices.