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Homework answers / question archive / Sanford-Brown College - MATH 2014 CHAPTER 16 SECTION 1-2: SIMPLE LINEAR REGRESSION AND CORRELATION 1)An inverse relationship between an independent variable x and a dependent variably y means that as x increases, y decreases, and vice versa
Sanford-Brown College - MATH 2014
CHAPTER 16 SECTION 1-2: SIMPLE LINEAR REGRESSION AND CORRELATION
1)An inverse relationship between an independent variable x and a dependent variably y means that as x increases, y decreases, and vice versa.
2. A direct relationship between an independent variable x and a dependent variably y means that the variables x and y increase or decrease together.
3. Another name for the residual term in a regression equation is random error.
4. A simple linear regression equation is given by
. The point estimate of y when x = 4 is 20.45.
5. The vertical spread of the data points about the regression line is measured by the y-intercept.
6. 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.
7. 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.
8. 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.
9. The residual ri is defined as the difference between the actual value yi and the estimated value
.
10. 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.
11. 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.
12. The residuals are observations of the error variable ?. Consequently, the minimized sum of squared deviations is called the sum of squares for error, denoted SSE.
13. Statisticians have shown that sample y-intercept b0 and sample slope coefficient b1 are unbiased estimators of the population regression parameters ?0 and ?1, respectively.
14. If cov(x, y) = 7.5075 and
, then the sample slope coefficient is 2.145.
15. The first-order linear model is sometimes called the simple linear regression model.
16. To create a deterministic model, we start with a probabilistic model that approximates the relationship we want to model.
MULTIPLE CHOICE
17. 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 |
18. 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. |
19. 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. |
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, 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. |
21. 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 |
22. 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. |
23. 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 |
24. 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 |
25. 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. |
26. 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 ?1 |
|
c. |
?0 and b1 |
|
d. |
?0 and ?1 |
27. 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. |
28. 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. |
29. 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 ?1 |
|
c. |
?0 and b1 |
|
d. |
?0 and ?1 |
30. 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 |
31. 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 |
32. 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. |
COMPLETION
33. In regression analysis, you predict the value of one variable on the basis of one or more other related variables. The variable being predicted is called the ____________________ variable, and the related variables used to make the prediction are called ____________________ variables.
34. A straight line regression model with only one independent variable is called a(n) ____________________-order linear model.
35. The objective of a regression model is to analyze the relationship between two variables, x and y, both of which must be based on ____________________ data.
36. The deviations between the actual data points and the fitted values from the model are called ____________________.
37. SSE stands for ____________________ of squares for ____________________.
38. In a simple linear regression model b1 is the ____________________ of the straight line.
39. In a simple linear regression model b0 is the ____________________ of the straight line.
40. Suppose the slope of a simple linear regression line between hours studying and exam score is 5. That means as ____________________ increases by one, ____________________ increases by 5.
41. You cannot interpret the ____________________ of the simple linear regression line unless the value of x = 0 lies within the range of where data was collected.
42. The method statisticians use to produce a straight line that minimizes the SSE is called the ____________________ method.
43. {Car Speed and Gas Mileage Narrative} Determine the least squares regression line.
44. {Car Speed and Gas Mileage Narrative} Estimate the gas mileage of a car traveling 70 mph.
45. The following 10 observations of variables x and y were collected.
|
x |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
y |
25 |
22 |
21 |
19 |
14 |
15 |
12 |
10 |
6 |
2 |
Find the least squares regression line, and the estimated value of y when x = 3.
46. A scatter diagram includes the following data points:
|
x |
3 |
2 |
5 |
4 |
5 |
|
y |
8 |
6 |
12 |
10 |
14 |
Two regression models are proposed: (1)
, and (2)
. Using the least squares method, which of these regression models provides the better fit to the data? Why?
47. Consider the following data values of variables x and y.
|
x |
2 |
4 |
6 |
8 |
10 |
13 |
|
y |
7 |
11 |
17 |
21 |
27 |
36 |
|
a. |
Determine the least squares regression line. |
|
b. |
Find the predicted value of y for x = 9. |
|
c. |
What does the value of the slope of the regression line tell you? |
48. {Sunshine and Melanoma Narrative} Determine the least squares regression line.
49. {Sunshine and Melanoma Narrative} Draw a scatter diagram of the data and plot the least squares regression line on it.
50. {Sunshine and Melanoma Narrative} Estimate the number of skin cancer cases per 100,000 people who live in a state that gets 6 hours of sunshine on average.
51. {Sunshine and Melanoma Narrative} What does the value of the slope of the regression line tell you?
52. {Sunshine and Melanoma Narrative} Calculate the residual corresponding to the pair (x, y) = (8, 15).
|
Salesperson |
Years of Experience |
Sales |
|
1 |
0 |
7 |
|
2 |
2 |
9 |
|
3 |
10 |
20 |
|
4 |
3 |
15 |
|
5 |
8 |
18 |
|
6 |
5 |
14 |
|
7 |
12 |
20 |
|
8 |
7 |
17 |
|
9 |
20 |
30 |
|
10 |
15 |
25 |
53. {Sales and Experience Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.
54. {Sales and Experience Narrative} Determine the least squares regression line.
55. {Sales and Experience Narrative} Interpret the value of the slope of the regression line.
56. {Sales and Experience Narrative} Estimate the monthly sales for a salesperson with 16 years of experience.
A professor of economics wants to study the relationship between income (y in $1000s) and education (x in years). A random sample eight individuals is taken and the results are shown below.
|
Education |
16 |
11 |
15 |
8 |
12 |
10 |
13 |
14 |
|
Income |
58 |
40 |
55 |
35 |
43 |
41 |
52 |
49 |
57. {Income and Education Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.
58. {Income and Education Narrative} Determine the least squares regression line.
59. {Income and Education Narrative} Interpret the value of the slope of the regression line.
60. {Income and Education Narrative} Estimate the income of an individual with 15 years of education.
An ardent fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last eight winners of her favorite game show. She records their winnings in dollars and the number of years of education. The results are as follows.
|
Contestant |
Years of Education |
Winnings |
|
1 |
11 |
750 |
|
2 |
15 |
400 |
|
3 |
12 |
600 |
|
4 |
16 |
350 |
|
5 |
11 |
800 |
|
6 |
16 |
300 |
|
7 |
13 |
650 |
|
8 |
14 |
400 |
61. {Trivia Games & Education Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.
62. {Trivia Games & Education Narrative} Determine the least squares regression line.
63. {Trivia Games & Education Narrative} Interpret the value of the slope of the regression line.
64. {Trivia Games & Education Narrative} Estimate the game winnings for a contestant with 15 years of education.
A financier whose specialty is investing in stage productions has observed that, in general, movies with "big-name" stars seem to generate more revenue than those plays whose stars are less well known. To examine his belief he records the gross revenue and the payment (in $ millions) given to the two highest-paid performers in the play for ten recently staged plays.
|
Play |
Cost of Two Highest Paid |
Gross Revenue |
|
|
Performers ($mil) |
($mil) |
|
1 |
5.3 |
48 |
|
2 |
7.2 |
65 |
|
3 |
1.3 |
18 |
|
4 |
1.8 |
20 |
|
5 |
3.5 |
31 |
|
6 |
2.6 |
26 |
|
7 |
8.0 |
73 |
|
8 |
2.4 |
23 |
|
9 |
4.5 |
39 |
|
10 |
6.7 |
58 |
65. {Theatre Revenues Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.
66. {Theatre Revenues Narrative} Determine the least squares regression line.
67. {Theatre Revenues Narrative} Interpret the value of the slope of the regression line.
68. {Theatre Revenues Narrative} Estimate the gross revenue of a play if the two highest paid performers received 6 million dollars.
69. {Theatre Revenues Narrative} Are the two highest paid performers worth all the money paid for them? Comment using the statistical analyses you have done.
The editor of a higher education book publisher claims that a large part of the cost of books is the cost of paper. This implies that larger textbooks will cost more money. As an experiment to analyze the claim, a university student visits the bookstore and records the number of pages and the selling price of twelve randomly selected textbooks. These data are listed below.
|
Textbook |
Number of Pages |
Selling Price ($) |
|
1 |
844 |
55 |
|
2 |
727 |
50 |
|
3 |
360 |
35 |
|
4 |
915 |
60 |
|
5 |
295 |
30 |
|
6 |
706 |
50 |
|
7 |
410 |
40 |
|
8 |
905 |
53 |
|
9 |
1058 |
65 |
|
10 |
865 |
54 |
|
11 |
677 |
42 |
|
12 |
912 |
58 |
70. {Cost of Textbooks Narrative} Determine the least squares regression line.
71. {Cost of Textbooks Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate.
72. {Cost of Textbooks Narrative} Interpret the value of the slope of the regression line.
73. {Cost of Textbooks Narrative} Estimate the selling price for a 650 pages book.
A statistician investigating the relationship between the amount of rain (in inches) and the number of automobile accidents gathered data on accidents in her city for 10 randomly selected days throughout the year. The results are shown below.
|
Day |
Rain |
Number of Accidents |
|
1 |
0.05 |
5 |
|
2 |
0.12 |
6 |
|
3 |
0.05 |
2 |
|
4 |
0.08 |
4 |
|
5 |
0.10 |
8 |
|
6 |
0.35 |
14 |
|
7 |
0.15 |
7 |
|
8 |
0.30 |
13 |
|
9 |
0.10 |
7 |
|
10 |
0.20 |
10 |
74. {Accidents and Rain Narrative} Find the least squares regression line.
75. {Accidents and Rain Narrative} Estimate the number of accidents in a day with 0.25 inches of rain.
76. {Accidents and Rain Narrative} What does the slope of the least squares regression line tell you?
77. {Accidents and Rain Narrative} What other variables might be associated with accidents, besides or along with rain?
Allman Brothers Concert
At a recent Allman Brothers concert, a survey was conducted that asked a random sample of 20 people their age and how many concerts they have attended since the first of the year. The following data were collected:
|
Age |
62 |
57 |
40 |
49 |
67 |
54 |
43 |
65 |
54 |
41 |
|
Number of Concerts |
6 |
5 |
4 |
3 |
5 |
5 |
2 |
6 |
3 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Age |
44 |
48 |
55 |
60 |
59 |
63 |
69 |
40 |
38 |
52 |
|
Number of Concerts |
3 |
2 |
4 |
5 |
4 |
5 |
4 |
2 |
1 |
3 |
An Excel output follows:
78. {Allman Brothers Concert Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate to describe the relationship between the age and number of concerts attended by the respondents.
79. {Allman Brothers Concert Narrative} Determine the least squares regression line.
80. {Allman Brothers Concert Narrative} Plot the least squares regression line on the scatter diagram.
81. {Allman Brothers Concert Narrative} Interpret the value of the slope of the regression line.
82. {Allman Brothers Concert Narrative} Estimate the number of Allman Brothers concerts attended by a 64 year old person.
Oil Quality and Price
Quality of oil is measured in API gravity degrees--the higher the degrees API, the higher the quality. The table shown below is produced by an expert in the field who believes that there is a relationship between quality and price per barrel.
|
Oil degrees API |
Price per barrel (in $) |
|
27.0 |
12.02 |
|
28.5 |
12.04 |
|
30.8 |
12.32 |
|
31.3 |
12.27 |
|
31.9 |
12.49 |
|
34.5 |
12.70 |
|
34.0 |
12.80 |
|
34.7 |
13.00 |
|
37.0 |
13.00 |
|
41.0 |
13.17 |
|
41.0 |
13.19 |
|
38.8 |
13.22 |
|
39.3 |
13.27 |
A partial Minitab output follows:
|
Descriptive Statistics |
||||
|
Variable |
N |
Mean |
StDev |
SE Mean |
|
Degrees |
13 |
34.60 |
4.613 |
1.280 |
|
Price |
13 |
12.730 |
0.457 |
0.127 |
|
Covariances |
||
|
|
Degrees |
Price |
|
Degrees |
21.281667 |
|
|
Price |
2.026750 |
0.208833 |
|
Regression Analysis |
|||||
|
Predictor |
Coef |
StDev |
T |
P |
|
|
Constant |
9.4349 |
0.2867 |
32.91 |
0.000 |
|
|
Degrees |
0.095235 |
0.008220 |
11.59 |
0.000 |
|
|
|
|
|
|
|
|
|
S = 0.1314 |
R-Sq = 92.46% |
R-Sq(adj) = 91.7% |
|||
|
Analysis of Variance |
|||||
|
Source |
DF |
SS |
MS |
F |
P |
|
Regression |
1 |
2.3162 |
2.3162 |
134.24 |
0.000 |
|
Residual Error |
11 |
0.1898 |
0.0173 |
|
|
|
Total |
12 |
2.5060 |
|
|
|
83. {Oil Quality and Price Narrative} Draw a scatter diagram of the data. Comment on whether it appears that a linear model might be appropriate to describe the relationship between the quality of oil and price per barrel.
84. {Oil Quality and Price Narrative} Determine the least squares regression line.
85. {Oil Quality and Price Narrative} Plot the least squares regression line on the scatter diagram.
86. {Oil Quality and Price Narrative} Interpret the value of the slope of the regression line.
87. {Oil Quality and Price Narrative} For what values of API gravity degrees do we feel comfortable making predictions for oil price?
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