Homework

12.1 Linear Equations

58.

For each of the following situations, state the independent variable and the dependent variable.

  1. A study is done to determine whether elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared with the age of drivers.
  2. A study is done to determine whether the weekly grocery bill changes based on the number of family members.
  3. Insurance companies base life insurance premiums partially on the age of the applicant.
  4. Utility bills vary according to power consumption.
  5. A study is done to determine whether a higher education reduces the crime rate in a population.
59.

Piece-rate systems are widely debated incentive payment plans. In a recent study of loan officer effectiveness, the following piece-rate system was examined:

 
% of goal reached < 80 80 100 120
Incentive n/a $4,000, with an additional $125 added per percentage point from 81% to 99% $6,500, with an additional $125 added per percentage point from 101% to 119% $9,500, with an additional $125 added per percentage point starting at 121%
Table 12.20

If a loan officer makes 95 percent of his or her goal, write the linear function that applies based on the incentive plan table. In context, explain the y-intercept and slope.

12.2 Scatter Plots

60.

The gross domestic product purchasing power parity (PPP) is an indication of a country’s currency value compared to another country. Table 12.21 shows the GDP PPP of Cuba compared with that of the United States. Construct a scatter plot of the data.

 
Year Cuba’s PPP Year Cuba’s PPP
1999 1,700 2006 4,000
2000 1,700 2007 11,000
2002 2,300 2008 9,500
2003 2,900 2009 9,700
2004 3,000 2010 9,900
2005 3,500    
Table 12.21
61.

The following table shows poverty rates and cell phone use in the United States. Construct a scatter plot of the data,

Year Poverty Rate Cellular Usage per Capita
2003 12.7 54.67
2005 12.6 74.19
2007 12 84.86
2009 12 90.82
Table 12.22
62.

Does the higher cost of tuition translate to higher-paying jobs? The table lists the top 10 colleges based on mid-career salary and the associated yearly tuition costs. Construct a scatter plot of the data.

 
School Mid-Career Salary (in thousands of U.S. dollars) Yearly Tuition (in U.S. dollars)
Princeton 137 28,540
Harvey Mudd 135 40,133
CalTech 127 39,900
U.S. Naval Academy 122 0
West Point 120 0
MIT 118 42,050
Lehigh University 118 43,220
NYU-Poly 117 39,565
Babson College 117 40,400
Stanford 114 54,506
Table 12.23
63.

If the level of significance is 0.05 and the p-value is 0.06, what conclusion can you draw?

64.

If there are 15 data points in a set of data, what is the number of degrees of freedom?

12.3 The Regression Equation

65.

What is the process through which we can calculate a line that goes through a scatter plot with a linear pattern?

66.

Explain what it means when a correlation has an r2-value of .72.

67.

Can a coefficient of determination be negative? Why or why not?

68.

The table below represents the relationship between SAT scores on the math portion of the test and high school grade point averages (GPAs).

Use the median–median line approach to find the equation for the line of best fit.

x (SAT math scores) y (GPAs)
624 90
544 86
363 70
373 71
350 65
741 98
262 60
587 87
327 62
364 67
261 50
Table 12.24

12.5 Prediction (Optional)

69.

Recently, the annual numbers of driver deaths per 100,000 people for the selected age groups are as follows:

Age (years) Number of Driver Deaths (per 100,000 people)
16–19 38
20–24 36
25–34 24
35–54 20
55–74 18
75+ 28
Table 12.25
  1. For each age group, pick the midpoint of the interval for the x value. For the 75+ group, use 80.
  2. Using age as the independent variable and number of driver deaths per 100,000 people as the dependent variable, make a scatter plot of the data.
  3. Calculate the least-squares (best–fit) line. Put the equation in the form ŷ = a + bx.
  4. Find the correlation coefficient. Is it significant?
  5. Predict the number of deaths for ages 40 years and 60 years.
  6. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?
  7. What is the slope of the least-squares (best-fit) line? Interpret the slope.
70.

Table 12.26 shows the life expectancy for an individual born in the United States in certain years.

Year of Birth Life Expectancy in years
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 12.26
  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Draw a scatter plot of the ordered pairs.
  3. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
  4. Find the correlation coefficient. Is it significant?
  5. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
  6. Why aren’t the answers to Part E the same as the values in Table 12.26 that correspond to those years?
  7. Use the two points in Part E to plot the least-squares line on your graph from Part B.
  8. Based on the data, is there a linear relationship between the year of birth and life expectancy?
  9. Are there any outliers in the data?
  10. 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.
  11. What is the slope of the least-squares (best-fit) line? Interpret the slope.
71.

The maximum discount value of the Entertainment® card for the Fine Dining section, 10th edition, for various pages is given in Table 12.27.

Page Number Maximum Value ($)
4 16
14 19
25 15
32 17
43 19
57 15
72 16
85 15
90 17
Table 12.27
  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Draw a scatter plot of the ordered pairs.
  3. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
  4. Find the correlation coefficient. Is it significant?
  5. Find the estimated maximum values for the restaurants on page 10 and on page 70.
  6. 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?
  7. Suppose there are 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?
  8. Is the least-squares line valid for page 200? Why or why not?
  9. What is the slope of the least-squares (best-fit) line? Interpret the slope.
72.

Table 12.28 gives the gold medal times for every other Summer Olympics for the women’s 100-meter freestyle in swimming.

Year Time in 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
Table 12.28
  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Draw a scatter plot of the data.
  3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  4. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
  5. Find the correlation coefficient. Is the decrease in times significant?
  6. Find the estimated gold medal time for 1932. Find the estimated time for 1984.
  7. Why are the answers from Part F different from the chart values?
  8. Does it appear that a line is the best way to fit the data? Why or why not?
  9. Use the least-squares line to estimate the gold medal time for the next Summer Olympics. Do you think your answer is reasonable? Why or why not?
73.
State No. of Letters in Name Year Entered the Union Rank for Entering the Union Area in square miles
Alabama 7 1819 22 52,423
Colorado 8 1876 38 104,100
Hawaii 6 1959 50 10,932
Iowa 4 1846 29 56,276
Maryland 8 1788 7 12,407
Missouri 8 1821 24 69,709
New Jersey 9 1787 3 8,722
Ohio 4 1803 17 44,828
South Carolina 13 1788 8 32,008
Utah 4 1896 45 84,904
Wisconsin 9 1848 30 65,499
Table 12.29

We are interested in whether the number of letters in a state name depends on the year the state entered the Union.

  1. Decide which variable should be the independent variable and which should be the dependent variable.
  2. Draw a scatter plot of the data.
  3. Does it appear from inspection that there is a relationship between the variables? Why or why not?
  4. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
  5. Find the correlation coefficient. What does it imply about the significance of the relationship?
  6. Find the estimated number of letters (to the nearest integer) a state name would have if it entered the Union in 1900. Find the estimated number of letters a state name would have if it entered the Union in 1940.
  7. Does it appear that a line is the best way to fit the data? Why or why not?
  8. Use the least-squares line to estimate the number of letters for a new state that enters the Union this year. Can the least-squares line be used to predict it? Why or why not?

12.6 Outliers

74.

Given the information in Table 12.30, which represents the relationship between final exam math grades and final exam history grades, decide whether point (56, 95) is an influential point. Explain how you arrived at your decision. Show all work.

x (final exam math grades) y (final exam history grades)
54 60
56 68
77 82
74 78
63 69
51 55
88 97
72 77
69 78
56 95
Table 12.30
75.

In Table 12.31, the height (sidewalk to roof) of notable tall buildings in America is compared with the number of stories of the building (beginning at street level).

Height (in feet) Stories
1,050 57
428 28
362 26
529 40
790 60
401 22
380 38
1,454 110
1,127 100
700 46
Table 12.31
  1. Using stories as the independent variable and height as the dependent variable, make a scatter plot of the data.
  2. Does it appear from inspection that there is a relationship between the variables?
  3. Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
  4. Find the correlation coefficient. Is it significant?
  5. Find the estimated heights for a building that has 32 stories and for a building that has 94 stories.
  6. Based on the data in Table 12.31, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?
  7. Are there any outliers in the data? If so, which point(s)?
  8. What is the estimated height of a building with six stories? Does the least-squares line give an accurate estimate of height? Explain why or why not.
  9. Based on the least-squares line, adding an extra story is predicted to add about how many feet to a building?
  10. What is the slope of the least-squares (best-fit) line? Interpret the slope.
76.

Ornithologists (scientists who study birds) tag sparrow hawks in 13 different colonies to study their population. They gather data for the percentage of new sparrow hawks in each colony and the percentage of those that have returned from migration.

Percent return—74, 66, 81, 52, 73, 62, 52, 45, 62, 46, 60, 46, 38


 
Percent new—5, 6, 8, 11, 12, 15, 16, 17, 18, 18, 19, 20, 20
  1. Enter the data into a calculator and make a scatter plot.
  2. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from Part A.
  3. Explain what the slope and y-intercept of the regression line tell us.
  4. How well does the regression line fit the data? Explain your response.
  5. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
  6. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70 percent of the adults from the previous year have returned. What is the prediction?
77.

The following table shows data on average per capita coffee consumption and death rate from heart disease in a random sample of 10 countries.

 
Yearly Coffee Consumption (liters) 2.5 3.9 2.9 2.4 2.9 0.8 9.1 2.7 0.8 0.7
No. of Deaths from Heart Disease 221 167 131 191 220 297 71 172 211 300
Table 12.32
  1. Enter the data into a calculator and make a scatter plot.
  2. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from Part A.
  3. Explain what the slope and y-intercept of the regression line tell us.
  4. How well does the regression line fit the data? Explain your response.
  5. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
  6. Do the data provide convincing evidence that there is a linear relationship between the amount of coffee consumed and the heart disease death rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question.
78.

The following table consists of one student athlete’s time (in minutes) to swim 2,000 yards and the student’s heart rate (beats per minute) after swimming on a random sample of 10 days.

 
Swim Time Heart Rate
34.12 144
35.72 152
34.72 124
34.05 140
34.13 152
35.73 146
36.17 128
35.57 136
35.37 144
35.57 148
Table 12.33
  1. Enter the data into a calculator and make a scatter plot.
  2. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot from Part A.
  3. Explain what the slope and y-intercept of the regression line tell us.
  4. How well does the regression line fit the data? Explain your response.
  5. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
79.

A researcher is investigating whether population impacts homicide rate. He uses demographic data from Detroit, Michigan, to compare homicide rates and the population.

 
Population Size Homicide Rate per 100,000 People
558,724 8.6
538,584 8.9
519,171 8.52
500,457 8.89
482,418 13.07
465,029 14.57
448,267 21.36
432,109 28.03
416,533 31.49
401,518 37.39
387,046 46.26
373,095 47.24
359,647 52.33
Table 12.34
  1. Use a calculator to construct a scatter plot of the data. What is the independent variable? Why?
  2. Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot.
  3. Discuss what the following mean in context:
    1. The slope of the regression equation
    2. The y-intercept of the regression equation
    3. The correlation coefficient, r
    4. The coefficient of determination, r2.
  4. Do the data provide convincing evidence that there is a linear relationship between population size and homicide rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question.
80.
 
School Mid-Career Salary (in thousands of U.S. dollars) Yearly Tuition (in U.S. dollars)
Princeton 137 28,540
Harvey Mudd 135 40,133
CalTech 127 39,900
U.S. Naval Academy 122 0
West Point 120 0
MIT 118 42,050
Lehigh University 118 43,220
NYU-Poly 117 39,565
Babson College 117 40,400
Stanford 114 54,506
Table 12.35

Use the data in the Table 12.35 to determine the linear regression line equation with the outliers removed. Is there a linear correlation for the data set with outliers removed? Justify your answer.