Deck 12: A: linear Regression and Correlation

Question 1

In a simple linear regression , the following statistics are calculated from a sample of ten observations: = 2250, = 10, = 50, = 75. Which of the following values equals the least-squares estimates of the slope and y-intercept, respectively?

A) 1.5 and 0.5
B) 1.5 and 2.5
C) 2.5 and -5.0
D) 2.5 and 1.5

Answer 1

C

Question 2

Which of the following correctly describes a true regression line?

A) It is represented by the equation E(y) =.
B) It is a line calculated from sample data by the method of least squares.
C) The values ofandfound in the equation of the true regression line represent the line's slope and intercept, respectively.
D) The response y is related to the independent variable x.

Answer 2

A

Question 3

Which of the following correctly describes an estimated regression line?

A) It is a line calculated from census data by the method of least squares.
B) It might be represented by the equation.
C) The values of a and b found in the equation of the estimated regression line represent the line's slope and intercept, respectively.

Answer 3

B

Question 4

A regression analysis between weight (y in kg) and height (x in cm) resulted in the following least-squares line: = -20 + 0.5x. Taking this into consideration, if the height is increased by 1 cm, what does this imply about the change in the weight, on average?

A) The weight will increase by 1/2 kg.
B) The weight will increase by 6 kg.
C) The weight will decrease by 1/2 kg.
D) The weight will decrease by 20 kg.

Question 5

If an estimated regression line has a y-intercept of 10 and a slope of 4, then when x = 2 what is the actual value of y?

A) 18
B) 15
C) 14
D) y is unknown

Question 6

In a regression setting, which of the following is NOT an assumption about the random error, ?

A) The errors are linearly correlated.
B) The errors are normally distributed with a mean of 0 and a common variance.
C) The errors are independent in a probabilistic sense.

Question 7

If all of the values of an independent variable x are equal, then regressing a dependent variable y on this independent variable x will result in which of the following coefficients of determination ( )?

A) -1.3
B) -1.0
C) 0.0
D) 1.0

Question 8

In a simple linear regression analysis, if SSE = 27, Total SS = 63, then what is the approximate percentage of the variation in the dependent variable y that is explained by the independent variable x?

A) 27%
B) 43%
C) 57%
D) 63%

Question 9

If the sum of squares for error is equal to 0, then what must the coefficient of determination ( ) equal?

A) 1.5
B) 1.0
C) 0.0
D) -1.0

Question 10

Which of the following is a measure of how well an estimated regression line fits the sample data on which it is based, denoted by (and equal to the proportion of the total variation in the values of the dependent variable, y, that can be explained by the association of y with x as measured by the estimated regression line)?

A) the sample coefficient of variation
B) the sample coefficient of correlation
C) the sample coefficient of determination
D) the sample coefficient of non-determination

Question 11

In the simple linear regression model, what does the slope represent?

A) the value of y when x = 0
B) the average change in y per unit change in x
C) the value of x when y = 0
D) the average change in x per unit change in y

Question 12

A regression analysis between sales (y in $1000) and advertising (x in $100) resulted in the following least-squares line: = 82 + 7x. Given this information, if advertising costs were $900, what could we reasonably expect the amount of sales (in dollars) to be?

A) $6,382
B) $82,063
C) $88,300
D) $145,000

Question 13

In a simple linear regression analysis, which of the following best describes the standard error of the slope?

A) It is a measure of the amount of change in the dependent variable y for a one-unit change in the independent variable x.
B) It is a measure of the variation in the regression slope from sample to sample.
C) It is equal to the square root of the standard error of the estimate.

Question 14

Which of the following is NOT an assumption for the simple linear regression model?

A) The distribution of the error terms will be skewed to left or right, depending on the values of the dependent variable.
B) The error terms have equal variances for all values of the independent variable.
C) The error terms are independent of each other.
D) The mean of the dependent variable for all levels of the independent variable can be connected by a straight line.

Question 15

In regression analysis, what do the residuals represent?

A) the difference between the actual y values and their predicted values
B) the difference between the actual x values and their predicted values
C) the square root of the slope of the regression line
D) the change in y per unit change in x

Question 16

For the values of the coefficient of determination listed below, which one yields the greatest value of sum of squares for regression given that the total sum of squares is 200?

A) -0.90
B) 0.00
C) 0.90
D) 0.98

Question 17

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least-squares line: = 75 +6x. From this information, if advertising is $800, then what is the predicted amount of sales (in dollars)?

A) $4,875
B) $12,300
C) $123,000
D) $487,500

Question 18

Given the least-squares regression line = 5 -2x, what may be said about the relationship between the two variables?

A) The relationship between x and y is positive.
B) The relationship between x and y is negative.
C) As x increases, so does y.
D) As x decreases, so does y.

Question 19

A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least-squares line: = 80,000 + 5x. What does this imply?

A) An increase of $1 in advertising is expected, on average, to result in an increase of $5 in sales.
B) An increase of $5 in advertising is expected, on average, to result in an increase of $5000 in sales.
C) An increase of $1 in advertising is expected, on average, to result in an increase of $5000 in sales.
D) An increase of $1 in advertising is expected, on average, to result in an increase of $80,005 in sales.

Question 20

In a simple linear regression , if the coefficient of determination is 0.96, what does this imply?

A) It means that 96% of the y values are positive.
B) It means that 90% of the total variation in y can be explained by the regression line.
C) It means that 96% of the x values are equal.
D) It means that 90% of the total variation in x can be explained by regression line.

Question 21

In a simple linear regression , the following sums of squares are produced: , , and . What percentage of the variation in y may be explained by the variation in x?

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

Question 22

Which of these coefficients of correlation (r) indicates a strong negative linear relationship between the two variables of interest?

A) -1.3
B) -0.9
C) 0.8
D) 0.9

Question 23

In regression analysis, if the coefficient of determination is 1.0, which of the following statements can be deduced from this information?

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 24

A regression line using 25 observations produced SSR = 118.68 and SSE = 56.32. What was the standard error of estimate?

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

Question 25

In simple linear regression, the plot of residuals versus fitted values can be used to check for which of the following

A) normality
B) a constant variance independent of x
C) independence

Question 26

In a regression the following pairs of (x, y) are given: (4, 1), (4, -1), (4, 0), (4, -2) and (4, 2). Which of the following statements may be deduced from the given information?

A) The correlation coefficient is -1.
B) The correlation coefficient is 0.
C) The correlation coefficient is 1.
D) The coefficient of determination is between -2 and 2.

Question 27

Which of the following is measured by the coefficient of determination ?

A) the amount of variation in y that is explained by variation in x
B) the amount of variation in x that is explained by variation in y
C) the amount of variation in y that is not explained by variation in x
D) the amount of variation in x that is not explained by variation in y

Question 28

In simple linear regression, most often we perform a two-tailed test of the population slope to determine whether there is sufficient evidence to infer that a linear relationship exists. How should we state the null hypothesis?

A)
B)
C)
D)

Question 29

In the least-squares regression line = 3 - 2x, which of the following is the correct predicted value of y?

A) 1.0 when x = 1.0
B) 1.0 when x = -1.0
C) 2.0 when x = 1.0
D) 2.0 when x = -1.0

Question 30

Given the least-squares regression line = -2.48 + 1.63x, and a coefficient of determination of 0.81, what is the coefficient of correlation?

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

Question 31

If the coefficient of correlation is 0.90, what is the percentage of the variation in the dependent variable y that is explained by the variation in the independent variable x?

A) 90%
B) 81%
C) 0.90%
D) 0.81%

Question 32

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

A) 0.300
B) 0.429
C) 0.700
D) 0.837

Question 33

Which of these coefficients of correlation (r) indicates a strong positive linear relationship between the two variables of interest?

A) -1.3
B) -0.9
C) 0.8
D) 0.9

Question 34

If the coefficient of determination is 0.975, then what may be said about the slope of the regression line?

A) It must be positive.
B) It must be negative.
C) It could be either positive or negative.

Question 35

What does the least-squares method for determining the best fit minimize?

A) total variation in the dependent variable
B) sum of squares for error
C) sum of squares for regression
D) total variation in the independent variable

Question 36

What is the symbol for the population coefficient of correlation?

A) r
B)
C) r²
D)

Question 37

Testing whether the slope of the population regression line could be 0 is equivalent to testing which of the following?

A) whether the sample coefficient of correlation could be 0
B) whether the standard error of estimate could be 0
C) whether the population coefficient of correlation could be 0
D) whether the sum of squares for error could be 0

Question 38

If a simple linear regression model has no y-intercept, then which of the following may be deduced about the values of the variables?

A) All values of x are 0.
B) All values of y are 0.
C) When y = 0, x = 0.
D) When x = 0, y = 0.

Question 39

Which value of the coefficient of correlation r indicates a stronger correlation than 0.65?

A) 0.60
B) 0.55
C) -0.45
D) -0.75

Question 40

In a simple linear regression including n = 10 observations, which of the following critical values would be appropriate for a 95% confidence interval estimation for the average value of y?

A) 1.860
B) 2.228
C) 2.262
D) 2.306

Question 41

Which of the following is the appropriate null hypothesis to test whether a population correlation is 0?

A)
B)
C)
D)

Question 42

The value of the sum of squares for regression can never be larger than 100.

Question 43

In order to predict with 90% confidence the expected value of y for a given value of x in a simple linear regression , a random sample of ten observations is taken. Which of the following t-table values would be used?

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

Question 44

Which of the following is an indication of no linear relationship between two variables x and y?

A) a coefficient of correlation of 1
B) a coefficient of correlation of 0
C) a coefficient of determination of -1
D) a coefficient of determination of 1

Question 45

In order to predict with 80% confidence the expected value of y for a given value of x in a simple linear regression , a random sample of 15 observations is taken. Which of the following t-table values would be used?

A) 1.350
B) 1.771
C) 2.160
D) 2.650

Question 46

In a simple linear regression , if the sum of squares for regression is 90, then the total sum of squares is at least 90.

Question 47

A study of 20 students showed that the correlation between the time spent writing a test and the number of hours studied the night before the test was 0.35. Using a level of significance equal to 0.05, what does this imply?

A) The sample correlation coefficient could be 0 since the test statistic does not fall into the rejection region.
B) The null hypothesis that the population mean is equal to 0 should not be rejected, and we should conclude that the true correlation coefficient is 0.
C) There is not enough statistical evidence to conclude that the true correlation coefficient is different from 0.
D) The null hypothesis that the population variance is equal to 0 should be rejected, and we should conclude that the true correlation coefficient is 0.

Question 48

In a simple linear regression , if the sum of squares for regression is 90, then the sum of squares for error is at most 90.

Question 49

A sample of 25 observations is selected, and the sample correlation coefficient between the variables x and y is r = 0.525. What is the test statistic value for testing vs.

A) about 3.81
B) about 3.65
C) about 3.08
D) about 2.96

Question 50

In a simple linear regression setting, the deterministic model equation determines an exact value of the dependent variable y when the value of the independent variable x is given, since all points must lie exactly on the line.

Question 51

In publishing the results of some research work, the following values of the correlation coefficient were listed. Which one is incorrect?

A) 0.00
B) 0.05
C) 0.95
D) 1.05

Question 52

In order to predict with 99% confidence the expected value of y for a given value of x in a simple linear regression , a random sample of ten observations is taken. Which of the following t-table values would be used?

A) 1.860
B) 2.306
C) 2.896
D) 3.355

Question 53

Given the least-squares regression line = -4.63 + 1.38x, and a coefficient of determination of 0.9025, what is the correlation coefficient?

A) -0.95
B) -0.81
C) +0.95
D) +1.38

Question 54

In order to predict with 98% confidence the expected value of y for a given value of x in a simple linear regression , a random sample of 15 observations is taken. Which of the following t-able values would be used?

A) 1.350
B) 1.771
C) 2.160
D) 2.650

Question 55

In a simple linear regression setting, the probabilistic model equation allows for some deviation of the points about the regression line, making it a more practical model.

Question 56

In a simple linear regression model, if the regression slope coefficient is negative, then the standard error of the estimate will be positive.

Question 57

Which of the following statements is NOT a property of the residuals in simple linear regression model?

A) They sum to 0.
B) They have a mean of 0..
C They have a median of 0
C) They have a standard deviation of 1.

Question 58

In a simple linear regression , if the sum of squares for regression is 90, then the correlation coefficient is 0.9.

Question 59

The sum of squares for regression can never be larger than the sum of squares for error.

Question 60

In a simple linear regression model, the regression slope coefficient will have the same sign as the correlation coefficient.

Question 61

The value of the sum of squares for error can never be larger than the total sum of squares.

Question 62

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 63

An automobile company in Ontario is interested in the relationship between the gender of its employees and employee productivity. A good starting point in this analysis would be to compute the coefficient of determination and the correlation coefficient.

Question 64

A regression analysis between sales (in $1000) and advertising (in $100) resulted in the following least-squares line: = 77 + 8x. This implies that if advertising is $600, then the predicted amount of sales is $125,000.

Question 65

The values of a and b found in the equation of the estimated regression line represent the line's y-intercept and slope, respectively, and are called estimated regression coefficients.

Question 66

In regression analysis, the independent variable is a variable whose value is known and is being used to explain or predict the value of another variable.

Question 67

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

Question 68

The values of α and found in the equation of the true regression line E(y) = represent the line's y-intercept and slope, respectively, and are called true regression coefficients.

Question 69

In simple linear regression, if the estimated values and the corresponding actual values are equal, then the standard error of estimate, SE( ), must equal -1.0.

Question 70

A regression analysis between sales (in $1000) and advertising (in $) resulted in the following least-squares line: = 60 + 5x. This implies that an increase of $1 in advertising is expected to result in an increase of $65 in sales.

Question 71

If a least-squares regression line has a y-intercept of 6.84 and a slope of 2.16, then when x = 1 the actual value of y must be 9.

Question 72

Regression analysis is a statistical method that seeks to establish an equation that allows the unknown value of one variable to be estimated from the known value of one or more other variables.

Question 73

A regression analysis between weight ( , in kg) and height ( , in cm) resulted in the following least-squares line: = -5 + 0.4 . This implies that if the height is increased by 1 cm, the weight is expected to increase by an average of 0.4 kg.

Question 74

If two variables are related in a negative linear manner, the scatterplot will show points on the x,y-space that are generally moving from the upper left to the lower right.

Question 75

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

Question 76

If the coefficient of determination is 0.982, then the slope of the regression line must be positive.

Question 77

In regression analysis, the dependent variable is a variable whose value is unknown and is being explained or predicted with the help of another variable.

Question 78

Given that the sum of squares for error is 52 and the sum of squares for regression is 148, then the coefficient of determination is 0.74.

Question 79

A measure of how well an estimated regression line fits the sample data on which it is based (denoted by and equal to the proportion of the total variation in the values of the dependent variable, y, that can be explained by the association of y with x as measured by the estimated regression line) is called the sample coefficient of correlation.

Question 80

One way to measure the strength of the relationship between the response variable y and the predictor variable x is to calculate the coefficient of determination, that is, the proportion of the total variation in y that is explained by the linear regression of y on x.