Deck 15: Simple Linear Regression and Correlation

Question 1

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.

Answer 1

All of these choices are correct.

Question 2

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.

Answer 2

0.736.

Question 3

Given that $s _ { x } ^ { 2 }$ = 500, $s _ { y } ^ { 2 }$ = 750, $s _ { x y }$ = 100 and n = 6, the standard error of estimate is:

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

Answer 3

30.2076.

Question 4

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. Which of the following best describes the null and alternative hypotheses needed for a test of significance?

A)Ho: $\beta$ 0 = 0 HA: $\beta$ 0 $\neq$ 0
B)Ho: $\beta$ 1 $\neq$ 0 HA: $\beta$ 1 = 0
C)Ho: $\beta$ 0 = 0 HA: $\beta$ 0 $\neq$ 1
D)Ho: $\beta$ 1 = 0 HA: $\beta$ 1 $\neq$ 0

Question 5

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.

Question 6

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 7

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 8

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 9

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 10

The symbol for the population coefficient of correlation is:

A)r.
B) $\rho$ .
C)r $2$ .
D) $\rho ^ { 2 }$ .

Question 11

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 12

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 13

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 14

The symbol for the sample coefficient of correlation is:

A)r.
B) $\rho$ .
C) $r ^ { 2 }$ .
D) $\rho ^ { 2 }$ .

Question 15

The following sums of squares are produced: $\Sigma$ $$\left( y _ { i } - \bar { y } \right) ^ { 2 }$$ = 250, $\Sigma$ $\left( y _ { i } - \hat { y } _ { i } \right) ^ { 2 }$ = 100, $\Sigma$ $\left( \hat { y } _ { i } - \bar { y } \right) ^ { 2 }$ = 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 16

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 17

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 18

The regression line Estimated y = 3 + 2x 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 19

Given the least squares regression line $\hat { y }$ = 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 20

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

A)Slope must be close to 1.
B)Slope must be close to -1.
C)Slope must be 0.95.
D)None of the above.

Question 21

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 $\hat { y }$ = 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 22

In testing the hypotheses: H0: $\rho$ s = 0
HA: $\rho$ s $\neq$ 0
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 23

A regression analysis between sales (in $1000) and advertising (in $) yielded the 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 $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 24

If the standard error of estimate = 15 and n = 12, then the sum of squares for error, SSE, is:

A)150.
B)180.
C)225.
D)2250.

Question 25

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 26

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)1 and 0.

Question 27

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 28

The smallest value that the standard error of estimatecan assume is:

A)-1.
B)0.
C)1.
D)-2.

Question 29

The standard error of estimate, , is given by:

A)SSE/(n - 2).
B) $\sqrt { S S E } / ( n - 2 )$ .
C) $\sqrt { S S E / ( n - 2 ) }$ .
D)SSE/ $\sqrt { n - 2 }$ .

Question 30

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 31

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 32

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 33

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 34

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 35

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 36

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 37

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 38

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 39

Of the values of the coefficient of determination listed below, which one implies the greatest value of the sum of squares for regression, given that the total variation in y is 1800?

A)0.69.
B)0.96.
C)96.
D)-100.

Question 40

A regression analysis between sales (in $1000) and advertising (in $100) yielded the least squares line $\hat { y }$ = 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 41

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 42

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

A) $\hat { \beta } _ { 0 }$ and $\hat { \beta } _ { 1 }$ .
B) $\hat { \beta } _ { 0 }$ and $\beta _ { 1 }$ .
C) $\beta _ { 0 }$ and $\hat { \beta } _ { 1 }$ .
D) $\beta _ { 0 }$ and $\beta _ { 1 }$ .

Question 43

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 44

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

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

Question 45

In regression analysis, the coefficient of determination, R2, 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 46

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

A) $\hat { \beta } _ { 0 }$ and $\hat { \beta } _ { 1 }$ .
B) $\hat { \beta } _ { 0 }$ and $\beta _ { 1 }$ .
C) $\beta _ { 0 }$ and $\hat { \beta } _ { 1 }$ .
D) $\beta _ { 0 }$ and $\beta _ { 1 }$ .

Question 47

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 $\bar { y }$ .
D)variation of x around the mean $\bar { x }$ .

Question 48

If cov(x,y) = -350, $s _ { x } ^ { 2 }$ = 900 and $s _ { y } ^ { 2 }$ = 225, then the coefficient of correlation is:

A)0.8819.
B)0.7778.
C)-0.0017.
D)0.0017.

Question 49

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 50

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 51

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 $\beta _ { 1 }$
B)Conduct a t-test for $\rho$ .
C)Conduct a t-test for $\beta _ { 1 }$ or a t-test for $\beta$ o.
D)Conduct a t-test for $\beta _ { 1 }$ or a t-test for $\rho$ .

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

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 54

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 55

When the sample size n is greater than 30, the Spearman rank correlation coefficient $r _ { s }$ is approximately normally distributed with:

A)mean 0 and standard deviation 1.
B)mean 1 and standard deviation $\sqrt { n - 1 }$ .
C)mean 1 and standard deviation 1/ $\sqrt { n - 1 }$ .
D)mean 0 and standard deviation 1/ $\sqrt { n - 1 }$ .

Question 56

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

A)Must be the same size and sign.
B)May 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 57

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 58

The least squares method requires that the variance $\sigma _ { \varepsilon } ^ { 2 }$ of the error variable $\varepsilon$ 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 59

When the variance, $\sigma _ { \varepsilon } ^ { 2 }$ , of the error variable $\varepsilon$ is a constant no matter what the value of x is, this condition is called:

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

Question 60

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 61

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

Question 62

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 63

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 64

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 65

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

Question 66

When the actual values y of a dependent variable and the corresponding predicted values $\ddot { p }$ are the same, the standard error of estimate, $S _ { \varepsilon }$ , will be 0.0.

Question 67

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

Question 68

The variance of the error variable, $\sigma _ { \varepsilon } ^ { 2 }$ , is required to be constant. When this requirement is violated, the condition is called heteroscedasticity.

Question 69

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 70

The variance of the error variable, $\sigma _ { \varepsilon } ^ { 2 }$ , is required to be constant. When this requirement is satisfied, the condition is called homoscedasticity.

Question 71

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

Question 72

In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

Question 73

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 74

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

Question 75

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 76

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

Question 77

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 78

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

Question 79

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

Question 80

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