100,000
for
the
selected
age
groups
was
as
follows
(Source:
http://
http://www.census.gov/compendia/statab/cats/transportation/motor_vehicle_accidents_and_fatalities.html 14
):
Age
Number of Driver Deaths per 100,000
16-19
38
20-24
36
25-34
24
35-54
20
55-74
18
75+
28
Table 12.5
a. For each age group, pick the midpoint of the interval for the x value. (For the 75+ group, use
80.)
b. Using “ages” as the independent variable and “Number of driver deaths per 100,000” as the
dependent variable, make a scatter plot of the data.
^
c. Calculate the least squares (best–fit) line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. Pick two ages and find the estimated fatality rates.
f. Use the two points in (e) to plot the least squares line on your graph from (b).
g. Based on the above data, is there a linear relationship between age of a driver and driver fatality
rate?
h. What is the slope of the least squares (best-fit) line? Interpret the slope.
13This content is available online at <http://cnx.org/content/m17085/1.14/>.
14http://www.census.gov/compendia/statab/cats/transportation/motor_vehicle_accidents_and_fatalities.html
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CHAPTER 12. LINEAR REGRESSION AND CORRELATION
Exercise 12.13.3
(Solution on p. 579.)
The average number of people in a family that received welfare for various years is given below.
(Source: House Ways and Means Committee, Health and Human Services Department )
Year
Welfare family size
1969
4.0
1973
3.6
1975
3.2
1979
3.0
1983
3.0
1988
3.0
1991
2.9
Table 12.6
a. Using “year” as the independent variable and “welfare family size” as the dependent variable,
make a scatter plot of the data.
^
b. Calculate the least squares line. Put the equation in the form of: y= a + bx
c. Find the correlation coefficient. Is it significant?
d. Pick two years between 1969 and 1991 and find the estimated welfare family sizes.
e. Use the two points in (d) to plot the least squares line on your graph from (b).
f. Based on the above data, is there a linear relationship between the year and the average number
of people in a welfare family?
g. Using the least squares line, estimate the welfare family sizes for 1960 and 1995. Does the least
squares line give an accurate estimate for those years? Explain why or why not.
h. Are there any outliers in the above data?
i. What is the estimated average welfare family size for 1986? Does the least squares line give an
accurate estimate for that year? Explain why or why not.
j. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.4
Use the AIDS data from the practice for this section (Section 12.12.2: Given), but this time use the
columns “year #” and “# new AIDS deaths in U.S.” Answer all of the questions from the practice
again, using the new columns.
Exercise 12.13.5
(Solution on p. 580.)
The height (sidewalk to roof) of notable tall buildings in America is compared to the number of
stories of the building (beginning at street level). (Source: Microsoft Bookshelf )
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557
Height (in feet)
Stories
1050
57
428
28
362
26
529
40
790
60
401
22
380
38
1454
110
1127
100
700
46
Table 12.7
a. Using “stories” as the independent variable and “height” as the dependent variable, make a
scatter plot of the data.
b. Does it appear from inspection that there is a relationship between the variables?
^
c. Calculate the least squares line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. Find the estimated heights for 32 stories and for 94 stories.
f. Use the two points in (e) to plot the least squares line on your graph from (b).
g. Based on the above data, is there a linear relationship between the number of stories in tall
buildings and the height of the buildings?
h. Are there any outliers in the above data? If so, which point(s)?
i. What is the estimated height of a building with 6 stories? Does the least squares line give an
accurate estimate of height? Explain why or why not.
j. Based on the least squares line, adding an extra story is predicted to add about how many feet
to a building?
k. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.6
Below is the life expectancy for an individual born in the United States in certain years. (Source:
National Center for Health Statistics)
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CHAPTER 12. LINEAR REGRESSION AND CORRELATION
Year of Birth
Life Expectancy
1930
59.7
1940
62.9
1950
70.2
1965
69.7
1973
71.4
1982
74.5
1987
75
1992
75.7
2010
78.7
Table 12.8
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Draw a scatter plot of the ordered pairs.
^
c. Calculate the least squares line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
f. Why aren’t the answers to part (e) the values on the above chart that correspond to those years?
g. Use the two points in (e) to plot the least squares line on your graph from (b).
h. Based on the above data, is there a linear relationship between the year of birth and life ex-
pectancy?
i. Are there any outliers in the above data?
j. Using the least squares line, find the estimated life expectancy for an individual born in 1850.
Does the least squares line give an accurate estimate for that year? Explain why or why not.
k. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.7
(Solution on p. 580.)
The percent of female wage and salary workers who are paid hourly rates is given below for the
years 1979 - 1992. (Source: Bureau of Labor Statistics, U.S. Dept. of Labor)
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559
Year
Percent of workers paid hourly rates
1979
61.2
1980
60.7
1981
61.3
1982
61.3
1983
61.8
1984
61.7
1985
61.8
1986
62.0
1987
62.7
1990
62.8
1992
62.9
Table 12.9
a. Using “year” as the independent variable and “percent” as the dependent variable, make a
scatter plot of the data.
b. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
c. Calculate the least squares line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. Find the estimated percents for 1991 and 1988.
f. Use the two points in (e) to plot the least squares line on your graph from (b).
g. Based on the above data, is there a linear relationship between the year and the percent of
female wage and salary earners who are paid hourly rates?
h. Are there any outliers in the above data?
i. What is the estimated percent for the year 2050? Does the least squares line give an accurate
estimate for that year? Explain why or why not?
j. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.8
The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition
10, for various pages is given below.
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CHAPTER 12. LINEAR REGRESSION AND CORRELATION
Page number
Maximum value ($)
4
16
14
19
25
15
32
17
43
19
57
15
72
16
85
15
90
17
Table 12.10
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Draw a scatter plot of the ordered pairs.
^
c. Calculate the least squares line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. Find the estimated maximum values for the restaurants on page 10 and on page 70.
f. Use the two points in (e) to plot the least squares line on your graph from (b).
g. Does it appear that the restaurants giving the maximum value are placed in the beginning of
the “Fine Dining” section? How did you arrive at your answer?
h. Suppose that there were 200 pages of restaurants. What do you estimate to be the maximum
value for a restaurant listed on page 200?
i. Is the least squares line valid for page 200? Why or why not?
j. What is the slope of the least squares (best-fit) line? Interpret the slope.
The next two questions refer to the following data: The cost of a leading liquid laundry detergent in
different sizes is given below.
Size (ounces)
Cost ($)
Cost per ounce
16
3.99
32
4.99
64
5.99
200
10.99
Table 12.11
Exercise 12.13.9
(Solution on p. 580.)
a. Using “size” as the independent variable and “cost” as the dependent variable, make a scatter
plot.
b. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
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561
^
c. Calculate the least squares line. Put the equation in the form of: y= a + bx
d. Find the correlation coefficient. Is it significant?
e. If the laundry detergent were sold in a 40 ounce size, find the estimated cost.
f. If the laundry detergent were sold in a 90 ounce size, find the estimated cost.
g. Use the two points in (e) and (f) to plot the least squares line on your graph from (a).
h. Does it appear that a line is the best way to fit the data? Why or why not?
i. Are there any outliers in the above data?
j. Is the least squares line valid for predicting what a 300 ounce size of the laundry detergent
would cost? Why or why not?
k. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.10
a. Complete the above table for the cost per ounce of the different sizes.
b. Using “Size” as the independent variable and “Cost per ounce” as the dependent variable,
make a scatter plot of the data.
c. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
d. Calculate the least squares line. Put the equation in the form of: y= a + bx
e. Find the correlation coefficient. Is it significant?
f. If the laundry detergent were sold in a 40 ounce size, find the estimated cost per ounce.
g. If the laundry detergent were sold in a 90 ounce size, find the estimated cost per ounce.
h. Use the two points in (f) and (g) to plot the least squares line on your graph from (b).
i. Does it appear that a line is the best way to fit the data? Why or why not?
j. Are there any outliers in the above data?
k. Is the least squares line valid for predicting what a 300 ounce size of the laundry detergent
would cost per ounce? Why or why not?
l. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.11
(Solution on p. 580.)
According to flyer by a Prudential Insurance Company representative, the costs of approximate
probate fees and taxes for selected net taxable estates are as follows:
Net Taxable Estate ($)
Approximate Probate Fees and Taxes ($)
600,000
30,000
750,000
92,500
1,000,000
203,000
1,500,000
438,000
2,000,000
688,000
2,500,000
1,037,000
3,000,000
1,350,000
Table 12.12
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Make a scatter plot of the data.
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CHAPTER 12. LINEAR REGRESSION AND CORRELATION
c. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
d. Calculate the least squares line. Put the equation in the form of: y= a + bx
e. Find the correlation coefficient. Is it significant?
f. Find the estimated total cost for a net taxable estate of $1,000,000. Find the cost for $2,500,000.
g. Use the two points in (f) to plot the least squares line on your graph from (b).
h. Does it appear that a line is the best way to fit the data? Why or why not?
i. Are there any outliers in the above data?
j. Based on the above, what would be the probate fees and taxes for an estate that does not have
any assets?
k. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.12
The following are advertised sale prices of color televisions at Anderson’s.
Size (inches)
Sale Price ($)
9
147
20
197
27
297
31
447
35
1177
40
2177
60
2497
Table 12.13
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Make a scatter plot of the data.
c. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
d. Calculate the least squares line. Put the equation in the form of: y= a + bx
e. Find the correlation coefficient. Is it significant?
f. Find the estimated sale price for a 32 inch television. Find the cost for a 50 inch television.
g. Use the two points in (f) to plot the least squares line on your graph from (b).
h. Does it appear that a line is the best way to fit the data? Why or why not?
i. Are there any outliers in the above data?
j. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.13
(Solution on p. 580.)
Below are the average heights for American boys. (Source: Physician’s Handbook, 1990 )
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563
Age (years)
Height (cm)
birth
50.8
2
83.8
3
91.4
5
106.6
7
119.3
10
137.1
14
157.5
Table 12.14
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Make a scatter plot of the data.
c. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
d. Calculate the least squares line. Put the equation in the form of: y= a + bx
e. Find the correlation coefficient. Is it significant?
f. Find the estimated average height for a one year–old. Find the estimated average height for an
eleven year–old.
g. Use the two points in (f) to plot the least squares line on your graph from (b).
h. Does it appear that a line is the best way to fit the data? Why or why not?
i. Are there any outliers in the above data?
j. Use the least squares line to estimate the average height for a sixty–two year–old man. Do you
think that your answer is reasonable? Why or why not?
k. What is the slope of the least squares (best-fit) line? Interpret the slope.
Exercise 12.13.14
The following chart gives the gold medal times for every other Summer Olympics for the women’s
100 meter freestyle (swimming).
Year
Time (seconds)
1912
82.2
1924
72.4
1932
66.8
1952
66.8
1960
61.2
1968
60.0
1976
55.65
1984
55.92
1992
54.64
2000
53.8
2008
53.1
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CHAPTER 12. LINEAR REGRESSION AND CORRELATION
Table 12.15
a. Decide which variable should be the independent variable and which should be the dependent
variable.
b. Make a scatter plot of the data.
c. Does it appear from inspection that there is a relationship between the variables? Why or why
not?
^
d. Calculate the least squares line. Put the equation in the form of: y= a + bx
e. Find the correlation coefficient. Is the decrease in times significant?
f. Find the estimated gold medal time for 1932. Find the estimated time for 1984.
g. Why are the answers from (f) different from the chart values?
h. Use the two points in (f) to plot the least squares line on your graph from (b).
i. Does it appear that a line is the best way to fit the data? Why or why not?
j. Use the least squares line to estimate the gold medal time for the next Summer Olympics. Do
you think that your answer is reasonable? Why or why not?
The next three questions use the following state information.
State
# letters in name
Year entered the
Rank for entering
Area
(square
Union
the Union
miles)
Alabama
7
1819
22
52,423
Colorado
1876
38
104,100
Hawaii
1959
50
10,932
Iowa
1846
29
56,276
Maryland
1788
7
12,407
Missouri
1821
24
69,709
New Jersey
1787
3
8,722
Ohio