Deck 16: Simple Linear Regression and Correlation

Question 1

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

Answer 1

True

Question 2

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

Answer 2

True

Question 3

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

Answer 3

True

Question 4

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

Question 5

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 6

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.

Question 7

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

Question 8

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 9

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

Question 10

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 11

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 12

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

Question 13

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 14

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

Question 15

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

Question 16

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

Question 17

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

Question 18

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 19

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 20

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 21

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

Question 22

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

Question 23

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

Question 24

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

Question 25

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 26

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

Question 27

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 28

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

Question 29

The probability distribution of the error variable $\varepsilon$ is normal, with mean E( $\varepsilon$ ) = 0, and standard deviation $\sigma$ _{$\varepsilon$} =1.

Question 30

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 31

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 32

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

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 _{$\varepsilon$} will be 0.0.

Question 34

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

Question 35

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

Question 36

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 37

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

Question 38

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 39

In simple linear regression, the denominator of the standard error of estimate s _{$\varepsilon$} is .

Question 40

If the error variable $\varepsilon$ is normally distributed, the test statistic for testing H₀: $\beta$ ₁ = 0 has a Student t-distribution with n - 2 degrees of freedom.

Question 41

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 42

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

Question 43

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 44

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 45

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 46

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

Question 47

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 48

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 49

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 50

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 51

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 52

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

Question 53

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 54

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 55

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 56

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 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

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 59

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 60

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 61

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 62

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 63

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.

Question 64

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

A) H₀: $\beta$ ₁ = 0
B) H₀: $\beta$ ₁ = b₁
C) H₀: $\beta$ ₁ $\neq$ 0
D) None of these choices.

Question 65

The symbol for the population coefficient of correlation is:

A) r
B) $\rho$
C) r²
D) $\rho$ ²

Question 66

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 67

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 68

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 69

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 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

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 72

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 73

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 74

Which of the following statistics and procedures can be used to determine whether a linear model should be employed?

A) The standard error of estimate.
B) The coefficient of determination.
C) The t-test of the slope.
D) All of these choices are true.

Question 75

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 76

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 77

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 78

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 79

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 80

The symbol for the sample coefficient of correlation is:

A) r
B) $\rho$
C) r²
D) $\rho$ ²