Deck 18: Simple Linear Regression and Correlation

Question 1

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.

Answer 1

C

Question 2

Which of the following best describes the value of the slope, if the coefficient of determination is 0.95?

A) Slope must be 1.
B) Slope must be negative.
C) Slope must be 0.95.
D) Slope must be positive.

Answer 2

D

Question 3

Which of the following best describes the relationship of the least squares regression line: Estimated y = 2 - x?

A) As x increases by 1 unit, y increases by 1 unit, estimated, on average.
B) As x increases by 1 unit y decreases by (2 -x) units, estimated, on average.
C) As x increases by 1 unit, y decreases by 1 unit, estimated, on average.
D) All of these choices are correct.

Answer 3

C

Question 4

Given that ss_x = 2500, ss_y = 3750, ss_xy = 500 and n = 6, the standard error of estimate is:

A) 12.247.
B) 24.933.
C) 30.2076.
D) 11.180.

Question 5

The Spearman rank correlation coefficient must be used to determine whether a relationship exists between two variables when:

A) one of the variables may be ordinal.
B) both of the variables may be ordinal.
C) both variables are interval and the normality requirement may not be satisfied.
D) All of these choices are correct.

Question 6

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

A) 2.1788.
B) 1.5648.
C) 1.5009.
D) 2.2716.

Question 7

Given that the sum of squares for error is 50 and the sum of squares for regression is 140, the coefficient of determination is:

A) 0.736.
B) 0.357.
C) 0.263.
D) 2.800.

Question 8

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 9

The following sums of squares are produced: (y_i-(y-bar))² = 250, 11ef1ab2_4bf1_c05d_a741_c73dacf30ae9_TB5762_11(y_i-(y_i-hat))² = 100, 11ef1ab2_4bf1_c05d_a741_c73dacf30ae9_TB5762_11((y_i-hat)-(y-bar))² = 150
The percentage of the variation in y that is explained by the variation in x is:

A) 60% .
B) 75% .
C) 40% .
D) 50% .

Question 10

Which value of the coefficient of correlation r indicates a stronger correlation than − 0.85?

A) −0.50
B) 0.75
C) −0.90
D) 0.85

Question 11

In a simple linear regression, which of the following is equivalent to testing the significance of the population slope?

A) Testing the significance of the population intercept.
B) Testing the significance of the mean.
C) Testing the significance of the coefficient of correlation.
D) All of these choices are correct.

Question 12

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

A) 80%.
B) 0.64%.
C) -80%.
D) 64%

Question 13

The symbol for the population coefficient of correlation is:

A) r.
B) .
C) r .
D) .

Question 14

If an estimated regression line has a y-intercept of 10 and a slope of -5, then when x = 0, the estimated value of y is:

A) −5
B) 0
C) 5
D) 10

Question 15

The symbol for the sample coefficient of correlation is:

A) r.
B) .
C) .
D) .

Question 16

Given a specific value of x and confidence level, which of the following statements is correct?

A) The confidence interval estimate of the expected value of y can be calculated but the prediction interval of y for the given value of x cannot be calculated.
B) The confidence interval estimate of the expected value of y will be wider than the prediction interval.
C) The prediction interval of y for the given value of x can be calculated but the confidence interval estimate of the expected value of y cannot be calculated.
D) The confidence interval estimate of the expected value of y will be narrower than the prediction interval.

Question 17

Given the least squares regression line y-hat = 3.52 - 1.27x, and a coefficient of determination of 0.81, the coefficient of correlation is:

A) -0.85.
B) 0.85.
C) -0.90.
D) 0.90.

Question 18

Which of the following statements best describes correlation analysis in a simple linear regression?

A) Correlation analysis measures the strength of a relationship between two numerical variables.
B) Correlation analysis measures the strength and direction of a linear relationship between two numerical variables.
C) Correlation analysis measures the direction of a relationship between two numerical variables.
D) Correlation analysis measures the strength of a relationship between two categorical variables.

Question 19

If the coefficient of correlation is −0.50, the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x is:

A) − 50%.
B) 25%.
C) 50%.
D) − 25%.

Question 20

In simple linear regression, most often we perform a two-tail test of the population slope to determine whether there is sufficient evidence to infer that a linear relationship exists. Which of the following best describes the null and alternative hypotheses needed for a test of significance?

A) Ho: β₀ = 0
H_A: β₀ ≠ 0
B) Ho: β₁ ≠ 0
H_A: β₁ = 0
C) Ho: β₀ = 0
H_A: β₀ ≠ 0
D) Ho: β₁ = 0
H_A: β₁ ≠ 0

Question 21

The standard error of estimate, , is given by:

A) SSE/(n - 2).
B) .
C) .
D) SSE/ .

Question 22

Which of the following best describes the coefficient of determination?

A) The coefficient of determination describes the percentage of variation in the x variable explained by the linear regression model on the y variable.
B) The coefficient of determination describes the direction of variation in the y variable explained by the linear regression model on the x variable.
C) The coefficient of determination describes the percentage of variation in the y variable explained by the linear regression model on the x variable.
D) The coefficient of determination describes the percentage of variation in the y variable unexplained by the linear regression model on the x variable.

Question 23

If the coefficient of determination is 81%, and the linear regression model has a negative slope, what is the value of the coefficient of correlation?

A) − 0.81
B) 0.90
C) 0.81
D) −0.90

Question 24

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 25

A regression analysis between sales (in $1000) and advertising (in $) yielded the least squares line
= 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 $5000 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 $5000 in sales.

Question 26

The standardised residual is defined as:

A) residual divided by the standard error of estimate.
B) residual multiplied by the square root of the standard error of estimate.
C) residual divided by the square of the standard error of estimate.
D) residual multiplied by the standard error of estimate.

Question 27

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

A) 1.0.
B) -1.0.
C) 0.0.
D) 2.0.

Question 28

The standard error of estimate, , is a measure of:

A) variation of y around the regression line.
B) variation of x around the regression line.
C) variation of y around the mean .
D) variation of x around the mean .

Question 29

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

A) the sum of squares for error is -1.0.
B) the sum of squares for regression is 1.0.
C) the sum of squares for error and sum of squares for regression are equal.
D) the sum of squares for regression and total variation in y are equal.

Question 30

A regression analysis between height y (in cm) and age x (in years) of 2 to 10 years old boys yielded the least squares line y-hat = 87 + 6.5x. This implies that by each additional year height is expected to:

A) increase by 93.5cm.
B) increase by 6.5cm.
C) increase by 87cm.
D) decrease by 6.5cm.

Question 31

Which of the following statements best describes why a linear regression is also called a least squares regression model?

A) A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the square of the difference between each actual x data value and the predicted x value.
B) A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the sum of the difference between each actual y data value and the predicted y value.
C) A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the square of each actual y data value and the predicted y value.
D) why a A linear regression is also called a least squares regression model because the regression line is calculated by minimizing the sum of the square of the differences between each actual y data value and the predicted y value.

Question 32

Given the data points (x,y) = (3,3), (4,4), (5,5), (6,6), (7,7), the least squares estimates of the y-intercept and slope are respectively:

A) 0 and 1.
B) -1 and 0.
C) 5 and 5.
D) 5 and 0.

Question 33

In testing the hypotheses: , the Spearman rank correlation coefficient in a sample of 50 observations is 0.389. The value of the test statistic is:

A) 2.75.
B) 18.178.
C) 2.723.
D) 17.995.

Question 34

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 35

Which of the following best describes if we want to test for a linear relationship between x and y, in regression analysis?

A) Conduct a t-test for
B) Conduct a t-test for .
C) Conduct a t-test for or a t-test for βo.
D) Conduct a t-test for or a t-test for .

Question 36

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line
= 75 +6x. This implies that if $800 is spent on advertising, then the predicted amount of sales (in dollars) is:

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

Question 37

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

A) and .
B) and .
C) and .
D) and .

Question 38

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 correct.

Question 39

Which of the following statements is correct when all the actual values of y are on an upward sloping regression line?

A) The coefficient of correlation is equal to one and the sign of the coefficient of correlation will be negative but the sign of the slope with be positive.
B) The coefficient of correlation and the slope must both be equal to 1.
C) The coefficient of correlation is equal to one and the sign of the coefficient of correlation and the sign of the slope will both be positive.
D) The coefficient of correlation and the slope must be equal to - 1.

Question 40

If the coefficient of correlation between x and y is close to −1.0, which of the following statements is correct?

A) There is a strong, positive linear relationship between x and y, where there may or may not be any causal relationship between x and y.
B) There is a strong, negative linear relationship between x and y, where there must be a causal relationship between x and y.
C) There is a weak, negative linear relationship between x and y, where there may or may not be any causal relationship between x and y.
D) There is a strong, negative linear relationship between x and y, where there may or may not be any causal relationship between x and y.

Question 41

The Pearson coefficient of correlation r equals 1 when there is/are no:

A) explained variation.
B) unexplained variation.
C) y-intercept in the model.
D) outliers.

Question 42

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) 225 and -1117.5.
B) 2.5 and -5.
C) 25 and-117.5.
D) 25 and 117.5.

Question 43

If a simple linear regression model has no y-intercept, then:

A) when x = 0, so does y.
B) y is always zero.
C) when y = 0, so does x.
D) y is always equal to x.

Question 44

In a regression problem the following pairs (x,y) are given: (1,2), (2.5,2), (3,2), (5,2) and (5.3,2). This indicates that the:

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

Question 45

Which of the following best describes the residuals in regression analysis?

A) The residuals are the difference between x data values observed and x predicted model values.
B) The residuals are the difference between the actual slope and the predicted model slope.
C) The residuals are the difference between the y data values observed and the predicted y model values.
D) The residuals are the addition of the y data values observed and predicted model values.

Question 46

When the sample size n is greater than 30, the Spearman rank correlation coefficient is approximately normally distributed with:

A) mean 0 and standard deviation 1.
B) mean 1 and standard deviation .
C) mean 1 and standard deviation 1/ .
D) mean 0 and standard deviation 1/ .

Question 47

Which of the following statements best describes the slope in the simple linear regression model?

A) The estimated average change in x per one unit increase in y.
B) The estimated average change in y when x = 1.
C) The estimated average value of y when x = 0.
D) The estimated average change in y per one unit increase in x.

Question 48

If the sum of squared residuals is zero, then the:

A) coefficient of determination must be 1.0.
B) coefficient of correlation must be 1.0.
C) coefficient of determination must be 0.0.
D) coefficient of correlation must be 0.0.

Question 49

In simple linear regression, which of the following statements indicates no linear relationship between the variables x and y?

A) The coefficient of determination is 1.0.
B) The coefficient of correlation is 0.0.
C) The sum of squares for error is 0.0.
D) The sum of squares for regression is relatively large.

Question 50

Which of the following best describes the y-intercept in the simple linear regression model?

A) The y-intercept is the estimated average value of y when x = 1.
B) The y-intercept is the estimated average value of x when y = 0.
C) The y-intercept is the rate of change of y with respect to changes in x.
D) The y-intercept is the estimated average value of y when x = 0.

Question 51

In simple linear regression, the coefficient of correlation r and the least squares estimate of the population slope :

A) Must be the same size and sign.
B) May have both have a different size and different sign.
C) May be the same size but have different sign.
D) May be different sizes but will have the same sign.

Question 52

On the least squares regression line Estimated y= 2 - 3x, the predicted value of y equals:

A) -1.0 when x = -1.0.
B) 1.0 when x = 1.0.
C) 5.0 when x = -1.0.
D) 5.0 when x = 1.0.

Question 53

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

A) and .
B) and .
C) and .
D) and .

Question 54

When the variance, , of the error variable is a constant no matter what the value of x is, this condition is called:

A) homocausality.
B) heteroscedasticity.
C) homoscedasticity.
D) heterocausality.

Question 55

In a regression problem, if all the values of the independent variable are equal, then the coefficient of determination must be:

A) 1.0.
B) 0.5.
C) 0.0.
D) -1.0.

Question 56

In regression analysis, the coefficient of determination, , measures the amount of variation in y that is:

A) caused by the variation in x.
B) explained by the variation in x.
C) unexplained by the variation in x.
D) caused by the variation in x or explained by the variation in x.

Question 57

The least squares method requires that the variance of the error variable is a constant no matter what the value of x is. When this requirement is violated, the condition is called:

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

Question 58

The least squares method for determining the best fit minimises:

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

Question 59

Which of the following is not a required condition for the error variable in the simple linear regression model?

A) The probability distribution of is normal.
B) The mean of the probability distribution of is zero.
C) The standard deviation of is a constant, no matter what the value of x.
D) The values of are auto correlated.

Question 60

In a regression problem, if the coefficient of determination is 0.95, this means that:

A) 95% of the y values are positive.
B) 95% of the variation in y can be explained by the variation in x.
C) 95% of the x values are equal.
D) 95% of the variation in x can be explained by the variation in y.

Question 61

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

Question 62

In developing a 90% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 15, the appropriate table value would be 1.761.

Question 63

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

Question 64

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 65

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 66

Regardless of the value of x, the standard deviation of the distribution of y values about the regression line is supposed to be constant. This assumption of equal standard deviations about the regression line is called multicollinearity.

Question 67

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 minimised.

Question 68

The variance of the error variable, , is required to be constant. When this requirement is satisfied, the condition is called homoscedasticity.

Question 69

The variance of the error variable, , is required to be constant. When this requirement is violated, the condition is called heteroscedasticity.

Question 70

In developing a 95% confidence interval for the expected value of y from a simple linear regression problem involving a sample of size 10, the appropriate table value would be 2.306.

Question 71

In simple linear regression, the divisor of the standard error of estimate, , is n - 1.

Question 72

A direct relationship between an independent variable x and a dependent variably y means that x and y move in the same directions.

Question 73

If all the values of an independent variable x are equal, then regressing a dependent variable y on x will result in a coefficient of determination of 100%.

Question 74

If there is no linear relationship between two variables and , the coefficient of correclation will be zero.

Question 75

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

Question 76

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

Question 77

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

Question 78

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 79

In a simple linear regression model, testing whether the slope, , of the population regression line is zero is the same as testing whether the population coefficient of correlation, , equals zero.

Question 80

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