# Chapter 12 The proportion of the variation in the dependent variable

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CHAPTER TWELVE
SIMPLE LINEAR REGRESSION
MULTIPLE-CHOICE QUESTIONS
In the following multiple-choice questions, circle the correct answer.
1. The proportion of the variation in the dependent variable y that is explained by the
estimated regression equation is measured by the
a. correlation coefficient
b. standard error of the estimate
c. coefficient of determination
d. confidence interval estimate
2. The least squares criterion is
a. min
()
2
ii
xy
b. min
()
2
i
yy
c. min
ˆ
()
2
ii
yy
d. min
ˆ
()
2
ii
yy
3. In a residual plot against x that does not suggest we should challenge the assumptions of
our regression model, we would expect to see
a. a horizontal band of points centered near zero
b. a widening band of points
c. a band of points having a slope consistent with that of the regression equation
d. a parabolic band of points
4. The difference between the observed value of the dependent variable and the value
predicted by using the estimated regression equation is the
a. standard error
b. residual
c. prediction interval
d. variance
5. As the goodness of fit for the estimated regression equation increases,
a. the absolute value of the regression equation’s slope increases
b. the value of the regression equation’s y intercept decreases
c. the value of the coefficient of determination increases
d. the value of the correlation coefficient increases
6. A measure of the strength of the relationship between two variables is the
a. coefficient of determination
b. slope b1 of the estimated regression line
c. standard error of the estimate
d. correlation coefficient
7. The interval estimate of the mean value of y for a given value of x is the
a. confidence interval
b. prediction interval
c. residual interval
d. correlation interval
8. Regression analysis is a statistical procedure for developing a mathematical equation that
describes how
a. one independent and one or more dependent variables are related
b. several independent and several dependent variables are related
c. one dependent and one or more independent variables are related
d. None of these answers is correct.
9. In regression analysis, the variable that is being predicted is the
a. dependent variable
b. independent variable
c. intervening variable
d. None of these answers is correct.
10. In a regression analysis, the variable that is being predicted
a. must have the same units as the variable doing the predicting
b. is the independent variable
c. is the dependent variable
d. usually is denoted by x
11. In regression analysis, the independent variable is
a. used to predict other independent variables
b. used to predict the dependent variable
c. called the intervening variable
d. None of these answers is correct.
12. The equation that describes how the dependent variable (y) is related to the independent
variable (x) is called
a. the correlation model
b. the regression model
c. correlation analysis
d. None of these answers is correct.
13. In a simple regression analysis (where y is a dependent and x an independent variable), if
the y intercept is positive, then
a. there is a positive correlation between x and y
b. there is a negative correlation between x and y
c. if x is increased, y must also increase
d. None of these answers is correct.
14. A procedure used for finding the equation of a straight line that provides the best
approximation for the relationship between the independent and dependent variables is
the
a. correlation analysis
b. mean squares method
c. least squares method
d. most squares method
15. Application of the least squares method results in values of the y intercept and the slope
that minimizes the sum of the squared deviations between the
a. observed values of the independent variable and the estimated values of the
independent variable
b. actual values of the independent variable and estimated values of the dependent
variable
c. observed values of the dependent variable and the estimated values of the
dependent variable
d. None of these answers is correct.
16. A least squares regression line
a. may be used to predict a value of y if the corresponding x value is given
b. implies a cause-effect relationship between x and y
c. can only be determined if a good linear relationship exists between x and y
d. All of these answers are correct.
17. In regression analysis if the dependent variable is measured in dollars, the independent
variable
a. must also be in dollars
b. must be in some unit of currency
c. can be any units
d. can not be in dollars
18. Regression analysis was applied between sales (in \$1,000) and advertising (in \$100), and
the following regression function was obtained.
ˆ
y
= 80 + 6.2x
Based on the above estimated regression line, if advertising is \$10,000, then the point
estimate for sales (in dollars) is
a. \$62,080
b. \$142,000
c. \$700
d. \$700,000
19. Regression analysis was applied between sales (in \$1000) and advertising (in \$100) and
the following regression function was obtained.
ˆ
y
= 500 + 4x
Based on the above estimated regression line if advertising is \$10,000, then the point
estimate for sales (in dollars) is
a. \$900
b. \$900,000
c. \$40,500
d. \$505,000
20. A regression analysis between demand (y in 1000 units) and price (x in dollars) resulted
in the following equation
ˆ
y
= 9 3x
The above equation implies that if the price is increased by \$1, the demand is expected to
a. increase by 6 units
b. decrease by 3 units
c. decrease by 6,000 units
d. decrease by 3,000 units
21. A regression analysis between sales (in \$1000) and price (in dollars) resulted in the
following equation
ˆ
y
= 50,000 8x
The above equation implies that an
a. increase of \$1 in price is associated with a decrease of \$8 in sales
b. increase of \$8 in price is associated with an increase of \$8,000 in sales
c. increase of \$1 in price is associated with a decrease of \$42,000 in sales
d. increase of \$1 in price is associated with a decrease of \$8000 in sales
22. A regression analysis between sales (y in \$1000) and advertising (x in dollars) resulted in
the following equation
ˆ
y
= 50,000 + 6x
The above equation implies that an
a. increase of \$6 in advertising is associated with an increase of \$6,000 in sales
b. increase of \$1 in advertising is associated with an increase of \$6 in sales
c. increase of \$1 in advertising is associated with an increase of \$56,000 in sales
d. increase of \$1 in advertising is associated with an increase of \$6,000 in sales
23. SSE can never be
a. larger than SST
b. smaller than SST
c. equal to 1
d. equal to zero
24. Which of the following is correct?
a. SSE = SSR + SST
b. SSR = SSE + SST
c. SST = SSR + SSE
d. SST = (SSR)2
25. In simple linear regression, r2 is the
a. estimated regression equation
b. coefficient of correlation
c. sum of the squared residuals
d. coefficient of determination
26. If all the points of a scatter diagram lie on the least squares regression line, then the
coefficient of determination for these variables based on this data is
a. 0
b. 1
c. either 1 or -1, depending upon whether the relationship is positive or negative
d. could be any value between -1 and 1
27. Larger values of r2 imply that the observations are more closely grouped about the
a. average value of the independent variables
b. average value of the dependent variable
c. least squares line
d. origin
28. In a regression analysis if r2 = 1, then
a. SSE must also be equal to one
b. SSE must be equal to zero
c. SSE can be any positive value
d. SSE must be negative
29. In a regression analysis if r2 = 1, then
a. SSE = SST
b. SSE = 1
c. SSR = SSE
d. SSR = SST
30. In regression and correlation analysis, if SSE and SST are known, then with this
information the
a. coefficient of determination can be computed
b. slope of the line can be computed
c. y intercept can be computed
d. All of the above can be computed.
31. In a regression analysis if SSE = 200 and SSR = 300, then the coefficient of
determination is
a. 0.6667
b. 0.6000
c. 0.4000
d. 1.5000
32. In a regression analysis if SSE = 500 and SSR = 300, then the coefficient of
determination is
a. 0.6000
b. 0.1666
c. 1.6666
d. 0.3750
33. If a data set has SST = 2,000 and SSE = 800, then the coefficient of determination is
a. 0.4
b. 0.6
c. 0.5
d. 0.8
EMBS4 TB 12 – 7
34. In a regression analysis if SST=4500 and SSE=1575, then the coefficient of
determination is
a. 0.35
b. 0.65
c. 2.85
d. 0.45
35. If the coefficient of determination is a positive value, then the regression equation
a. must have a positive slope
b. must have a negative slope
c. could have either a positive or a negative slope
d. must have a positive y intercept
36. It is possible for the coefficient of determination to be
a. larger than 1
b. less than one
c. less than zero
d. All of these answers are correct, depending on the situation under consideration.
37. If the coefficient of correlation is 0.8, the percentage of variation in the dependent
variable explained by the estimated regression equation is
a. 0.80%
b. 80%
c. 0.64%
d. 64%
38. If the coefficient of correlation is 0.4, the percentage of variation in the dependent
variable explained by the estimated regression equation
a. is 40%
b. is 16%
c. is 4%
d. can be any positive value
39. If the coefficient of determination is equal to 1, then the coefficient of correlation
a. must also be equal to 1
b. can be either -1 or +1
c. can be any value between -1 to +1
d. must be -1
40. If there is a very strong correlation between two variables, then the coefficient of
correlation must be
a. much larger than 1, if the correlation is positive
b. much smaller than 1, if the correlation is negative
c. either much larger than 1 or much smaller than 1
d. None of these answers is correct.
41. If the coefficient of correlation is a positive value, then the slope of the regression line
a. must also be positive
b. can be either negative or positive
c. can be zero
d. None of these answers is correct.
42. If the coefficient of correlation is a negative value, then the coefficient of determination
a. must also be negative
b. must be zero
c. can be either negative or positive
d. must be positive
43. The numerical value of the coefficient of determination
a. is always larger than the coefficient of correlation
b. is always smaller than the coefficient of correlation
c. is negative if the coefficient of determination is negative
d. can be larger or smaller than the coefficient of correlation
44. If the coefficient of determination is 0.81, the coefficient of correlation
a. is 0.6561
b. must be 0.9
c. must be positive
d. None of these answers is correct.
45. If two variables, x and y, have a strong linear relationship, then
a. there may or may not be any causal relationship between x and y
b. x causes y to happen
c. y causes x to happen
d. None of these answers is correct.
Exhibit 12-1
A regression analysis resulted in the following information regarding a dependent variable (y) and
an independent variable (x).
n = 10
x = 55
y = 55
x2 = 385
46. Refer to Exhibit 12-1. The least squares estimate of b1 equals
a. 1
b. -1
c. 5.5
d. 11
47. Refer to Exhibit 12-1. The least squares estimate of b0 equals
a. 1
b. -1
c. 5.5
d. 11
48. Refer to Exhibit 12-1. The point estimate of y when x = 20 is
a. 0
b. 31
c. 9
d. -9
49. Refer to Exhibit 12-1. The sample correlation coefficient equals
a. 0
b. -1
c. +1
d. -0.5
50. Refer to Exhibit 12-1. The coefficient of determination equals
a. 0
b. -1
c. +1
d. -0.5
Exhibit 12-2
You are given the following information about y and x.
y
x
Dependent
Variable
Independent
Variable
5
15
7
12
9
10
11
7
51. Refer to Exhibit 12-2. The least squares estimate of b1 equals
a. -0.7647
b. -0.13
c. 21.4
d. 16.412
52. Refer to Exhibit 12-2. The least squares estimate of b0 equals
a. -7.647
b. -1.3
c. 21.4
d. 16.41176
53. Refer to Exhibit 12-2. The sample correlation coefficient equals
a. -86.667
b. -0.99705
c. 0.9941
d. 0.99705
54. Refer to Exhibit 12-2. The coefficient of determination equals
a. -0.99705
b. -0.9941
c. 0.9941
d. 0.99705
55. In regression analysis, which of the following is not a required assumption about the error
term
?
a. The expected value of the error term is zero.
b. The variance of the error term is the same for all values of x.
c. The values of the error term are independent.
d. All are required assumptions about the error term.
56. In simple linear regression analysis, which of the following is not true?
a. The F test and the t test yield the same results.
b. The F test and the t test may or may not yield the same results.
c. The relationship between x and y is represented by means of a straight line.
d. The value of F = t2.
Exhibit 12-3
Regression analysis was applied between sales data (in \$1,000s) and advertising data (in \$100s)
and the following information was obtained.
y
= 12 + 1.8x
n = 17
SSR = 225
SSE = 75
sb1 = 0.2683
57. Refer to Exhibit 12-3. Based on the above estimated regression equation, if advertising is
\$3,000, then the point estimate for sales (in dollars) is
a. \$66,000
b. \$5,412
c. \$66
d. \$17,400
58. Refer to Exhibit 12-3. The F statistic computed from the above data is
a. 3
b. 45
c. 48
d. Not enough information is given to answer this question.
59. Refer to Exhibit 12-3. The critical F value at = 0.05 is
a. 3.59
b. 3.68
c. 4.45
d. 4.54
60. Refer to Exhibit 12-3. The t statistic for testing the significance of the slope is
a. 1.80
b. 1.96
c. 6.709
d. 0.555
61. Refer to Exhibit 12-3. Using = 0.05, the critical t value for testing the significance of
the slope is
a. 1.753
b. 2.131
c. 1.746
d. 2.120
62. Compared to the confidence interval estimate for a particular value of y (in a linear
regression model), the interval estimate for an average value of y will be
a. narrower
b. wider
c. the same
d. Not enough information is given.
63. A data point (observation) that does not fit the trend shown by the remaining data is
called a(n)
a. residual
b. outlier
c. point estimate
d. None of the other answers is correct.
64. An observation that has a strong effect on the regression results is called a(n)
a. residual
b. sum of squares error
c. influential observation
d. None of the other answers is correct.
65. Data points having high leverage are often
a. residuals
b. sum of squares error
c. influential
d. None of the other answers is correct.
PROBLEMS
1. Given below are seven observations collected in a regression study on two variables, x
(independent variable) and y (dependent variable).
x
y
2
12
3
9
6
8
7
7
8
6
7
5
9
2
a. Develop the least squares estimated regression equation.
b. At 95% confidence, perform a t test and determine whether or not the slope is
significantly different from zero.
c. Perform an F test to determine whether or not the model is significant. Let = 0.05.
d. Compute the coefficient of determination.
2. Given below are seven observations collected in a regression study on two variables, x
(independent variable) and y (dependent variable). Use Excel to develop a scatter
diagram and to compute the least squares estimated regression equation.
x
y
2
12
3
9
6
8
7
7
8
6
7
5
9
2
3. A company has recorded data on the daily demand for its product (y in thousands of
units) and the unit price (x in hundreds of dollars). A sample of 11 days demand and
associated price resulted in the following data.
x = 154
x 2 = 2,586
y =451
y 2 = 18,901
x y = 5,930
a. Using the above information, develop the least-squares estimated regression line.
b. Compute the coefficient of determination.

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