12.1 Linear Equations
For each of the following situations, state the independent variable and the dependent variable.
- 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.
- A study is done to determine whether the weekly grocery bill changes based on the number of family members.
- Insurance companies base life insurance premiums partially on the age of the applicant.
- Utility bills vary according to power consumption.
- A study is done to determine whether a higher education reduces the crime rate in a population.
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% |
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 The Regression Equation
What is the process through which we can calculate a line that goes through a scatter plot with a linear pattern?
Can a coefficient of determination be negative? Why or why not?
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 |
12.4 Prediction (Optional)
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 |
- For each age group, pick the midpoint of the interval for the x value. For the 75+ group, use 80.
- 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.
- Calculate the least-squares (best–fit) line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Predict the number of deaths for ages 40 years and 60 years.
- Based on the given data, is there a linear relationship between age of a driver and driver fatality rate?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
Table 12.21 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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated life expectancy for an individual born in 1950 and for one born in 1982.
- Why aren’t the answers to Part E the same as the values in Table 12.21 that correspond to those years?
- Use the two points in Part E to plot the least-squares line on your graph from Part B.
- Based on the data, is there a linear relationship between the year of birth and life expectancy?
- Are there any outliers in the data?
- 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.
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
The maximum discount value of the Entertainment® card for the Fine Dining section, 10th edition, for various pages is given in Table 12.22.
| Page Number | Maximum Value ($) |
|---|---|
| 4 | 16 |
| 14 | 19 |
| 25 | 15 |
| 32 | 17 |
| 43 | 19 |
| 57 | 15 |
| 72 | 16 |
| 85 | 15 |
| 90 | 17 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated maximum values for the restaurants on page 10 and on page 70.
- 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?
- Suppose there are 200 pages of restaurants. What do you estimate to be the maximum value for a restaurant listed on page 200?
- Is the least-squares line valid for page 200? Why or why not?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
Table 12.23 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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is the decrease in times significant?
- Find the estimated gold medal time for 1932. Find the estimated time for 1984.
- Why are the answers from Part F different from the chart values?
- Does it appear that a line is the best way to fit the data? Why or why not?
- 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?
| 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 |
We are interested in whether the number of letters in a state name depends on the year the state entered the Union.
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables? Why or why not?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. What does it imply about the significance of the relationship?
- 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.
- Does it appear that a line is the best way to fit the data? Why or why not?
- 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?
12.5 Outliers
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 |
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 |
- Using stories as the independent variable and height as the dependent variable, make a scatter plot of the data.
- Does it appear from inspection that there is a relationship between the variables?
- Calculate the least-squares line. Put the equation in the form ŷ = a + bx.
- Find the correlation coefficient. Is it significant?
- Find the estimated heights for a building that has 32 stories and for a building that has 94 stories.
- Based on the data in Table 12.26, is there a linear relationship between the number of stories in tall buildings and the height of the buildings?
- Are there any outliers in the data? If so, which point(s)?
- 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.
- Based on the least-squares line, adding an extra story is predicted to add about how many feet to a building?
- What is the slope of the least-squares (best-fit) line? Interpret the slope.
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
- Enter the data into a calculator and make a scatter plot.
- 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.
- Explain what the slope and y-intercept of the regression line tell us.
- How well does the regression line fit the data? Explain your response.
- Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
- 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?
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 |
- Enter the data into a calculator and make a scatter plot.
- 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.
- Explain what the slope and y-intercept of the regression line tell us.
- How well does the regression line fit the data? Explain your response.
- Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
- 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.
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 |
- Enter the data into a calculator and make a scatter plot.
- 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.
- Explain what the slope and y-intercept of the regression line tell us.
- How well does the regression line fit the data? Explain your response.
- Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain.
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 |
- Use a calculator to construct a scatter plot of the data. What is the independent variable? Why?
- Use the calculator’s regression function to find the equation of the least-squares regression line. Add this to your scatter plot.
- Discuss what the following mean in context:
- The slope of the regression equation
- The y-intercept of the regression equation
- The correlation coefficient, r
- The coefficient of determination, r2
- 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.
| 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 |
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.