12.8 Exercises
- What are the dependent and independent variables?
- Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
- Graph the equation from 2.
2. Is the equation [latex]y = 10 + 5x – 3x^2[/latex] linear? Why or why not?
3. Which of the following equations are linear?
- [latex]y = 6x + 8[/latex]
- [latex]y + 7 = 3x[/latex]
- [latex]y – x = 8x^2[/latex]
- [latex]4y= 8[/latex]
4. The table below contains real data for the first two decades of AIDS reporting. Use the columns “year” and “# AIDS cases diagnosed. Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?
Year | # AIDS cases diagnosed | # AIDS deaths |
Pre-1981 | 91 | 29 |
1981 | 319 | 121 |
1982 | 1,170 | 453 |
1983 | 3,076 | 1,482 |
1984 | 6,240 | 3,466 |
1985 | 11,776 | 6,878 |
1986 | 19,032 | 11,987 |
1987 | 28,564 | 16,162 |
1988 | 35,447 | 20,868 |
1989 | 42,674 | 27,591 |
1990 | 48,634 | 31,335 |
1991 | 59,660 | 36,560 |
1992 | 78,530 | 41,055 |
1993 | 78,834 | 44,730 |
1994 | 71,874 | 49,095 |
1995 | 68,505 | 49,456 |
1996 | 59,347 | 38,510 |
1997 | 47,149 | 20,736 |
1998 | 38,393 | 19,005 |
1999 | 25,174 | 18,454 |
2000 | 25,522 | 17,347 |
2001 | 25,643 | 17,402 |
2002 | 26,464 | 16,371 |
Total | 802,118 | 489,093 |
5. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is [latex]y = 50 + 100x[/latex].
- What are the independent and dependent variables?
- What is the y-intercept and what is the slope? Interpret them using complete sentences.
6. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is [latex]y = 12,000x[/latex].
- What are the independent and dependent variables?
- How many pounds of soil does the shoreline lose in a year?
- What is the y-intercept? Interpret its meaning.
7. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is [latex]y = 15 – 1.5x[/latex] where [latex]x[/latex] is the number of hours passed in an eight-hour day of trading.
- What are the slope and y-intercept? Interpret their meaning.
- If you owned this stock, would you want a positive or negative slope? Why?
8. For each of the following situations, state the independent variable and the dependent variable.
- A study is done to determine if elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers.
- A study is done to determine if 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 if a higher education reduces the crime rate in a population.
9. Does the scatter plot appear linear? Strong, moderate, or weak? Positive or negative?
10. Does the scatter plot appear linear? Strong, moderate, or weak? Positive or negative?
11. Does the scatter plot appear linear? Strong, moderate, or weak? Positive or negative?
12. The Gross Domestic Product Purchasing Power Parity is an indication of a country’s currency value compared to another country. The table below shows the GDP PPP of Cuba as compared to US dollars. 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 |
13. The following table shows the poverty rates and cell phone usage 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 |
14. Does the higher cost of tuition translate into higher-paying jobs? The table lists the top ten 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) | Yearly Tuition |
Princeton | 137 | 28,540 |
Harvey Mudd | 135 | 40,133 |
CalTech | 127 | 39,900 |
US 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 |
15. A random sample of ten professional athletes produced the following data where [latex]x[/latex] is the number of endorsements the player has and [latex]y[/latex] is the amount of money made (in millions of dollars).
[latex]x[/latex] | [latex]y[/latex] | [latex]x[/latex] | [latex]y[/latex] |
0 | 2 | 5 | 12 |
3 | 8 | 4 | 9 |
2 | 7 | 3 | 9 |
1 | 3 | 0 | 3 |
5 | 13 | 4 | 10 |
- Draw a scatter plot of the data.
- Use regression to find the equation for the line of best fit.
- Draw the line of best fit on the scatter plot.
- What is the slope of the line of best fit? What does it represent?
- What is the [latex]y[/latex]-intercept of the line of best fit? What does it represent?
16. What does an r value of zero mean?
17. What is the process through which we can calculate a line that goes through a scatter plot with a linear pattern?
18. Explain what it means when a correlation has an [latex]r^2[/latex] of 0.72.
19. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where [latex]x[/latex] is the day. The model can be written as [latex]\hat{y} = 101.32 + 2.48x[/latex] where [latex]\hat{y}[/latex] is in thousands of dollars.
- What would you predict the sales to be on day 60?
- What would you predict the sales to be on day 90?
20. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is [latex]\hat{y}= 1350 – 1.2x[/latex] where [latex]x[/latex] is the number of hours and [latex]\hat{y}[/latex] represents the number of acres left to mow.
- How many acres will be left to mow after 20 hours of work?
- How many acres will be left to mow after 100 hours of work?
- How many hours will it take to mow all of the lawns?
21. The table below contains real data for the first two decades of AIDS reporting.
Year | # AIDS cases diagnosed | # AIDS deaths |
Pre-1981 | 91 | 29 |
1981 | 319 | 121 |
1982 | 1,170 | 453 |
1983 | 3,076 | 1,482 |
1984 | 6,240 | 3,466 |
1985 | 11,776 | 6,878 |
1986 | 19,032 | 11,987 |
1987 | 28,564 | 16,162 |
1988 | 35,447 | 20,868 |
1989 | 42,674 | 27,591 |
1990 | 48,634 | 31,335 |
1991 | 59,660 | 36,560 |
1992 | 78,530 | 41,055 |
1993 | 78,834 | 44,730 |
1994 | 71,874 | 49,095 |
1995 | 68,505 | 49,456 |
1996 | 59,347 | 38,510 |
1997 | 47,149 | 20,736 |
1998 | 38,393 | 19,005 |
1999 | 25,174 | 18,454 |
2000 | 25,522 | 17,347 |
2001 | 25,643 | 17,402 |
2002 | 26,464 | 16,371 |
Total | 802,118 | 489,093 |
- Graph “year” versus “# AIDS cases diagnosed” (plot the scatter plot). Do not include pre-1981 data.
- Calculate the correlation coefficient.
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the predicted number of diagnosed cases for the year 1985?
- What is the predicted number of diagnosed cases for the year 1970? Why doesn’t this answer make sense?
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
22. Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows:
Age | Number of Driver Deaths per 100,000 |
17.5 | 38 |
22 | 36 |
29.5 | 24 |
44.5 | 20 |
64.5 | 18 |
80 | 28 |
- 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.
- Calculate the least squares (best–fit) line.
- Interpret the slope of the least squares line.
- Predict the number of deaths people aged 40.
- Find the correlation coefficient.
- Interpret the correlation coefficient.
- Find the coefficient of determination.
- Interpret the coefficient of determination.
- Find the standard error of the estimate.
- Interpret the standard error of the estimate.
23. The table below shows the life expectancy for an individual born in the United States in certain years.
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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated life expectancy for someone born in 1950? Why doesn’t this value match the life expectancy given in the table for 1950?
- What is the estimated life expectancy for someone born in 1982?
- Using the regression equation, find the estimated life expectancy for someone born in 1850. Is this an accurate estimate for that year? Explain why or why not.
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
24. The maximum discount value of the Entertainment® card for the “Fine Dining” section, Edition ten, for various pages is given in the table below.
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.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated maximum value for restaurants on page 10?
- What is the estimated maximum value for restaurants on page 70?
- Using the regression equation, find the estimated maximum value for restaurants on page 200. Is this an accurate estimate for that page? Explain why or why not.
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
25. The table below 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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated gold medal time for 1932?
- What is the estimated gold medal time for 1984?
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
26. 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).
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, draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated height for a 32 story building?
- What is the estimated height for a 94 story building?
- Using the regression equation, find the estimated height for a 6 story building. Is this an accurate estimate for the height of a 6 story building? Explain why or why not.
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate
27. The following table shows data on average per capita wine consumption and heart disease rate in a random sample of 10 countries.
Yearly wine consumption in liters | 2.5 | 3.9 | 2.9 | 2.4 | 2.9 | 0.8 | 9.1 | 2.7 | 0.8 | 0.7 |
Death from heart diseases | 221 | 167 | 131 | 191 | 220 | 297 | 71 | 172 | 211 | 300 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
28. The following table consists of one student athlete’s time (in minutes) to swim 2000 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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated heart rate for a swim time of 34.75 minutes?
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
29. The table below gives percent of workers who are paid hourly rates for the years 1979 to 1992.
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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated percent of workers paid hourly rates in 1988?
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
30. The table below shows the average heights for American boys in 1990.
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 |
- Decide which variable should be the independent variable and which should be the dependent variable.
- Draw a scatter plot of the ordered pairs.
- Find the correlation coefficient
- Interpret the correlation coefficient.
- Find the linear regression equation.
- Interpret the slope of the linear regression equation.
- What is the estimated average height for a one-year old?
- Using the regression equation, find the estimated average height for a 62 year old man. Do you think that your answer is reasonable? Explain why or why not.
- Calculate the coefficient of determination.
- Interpret the coefficient of determination.
- Calculate the standard error of the estimate.
- Interpret the standard error of the estimate.
Attribution
“Chapter 12 Homework” and “Chapter 12 Practice” in Introductory Statistics by OpenStax Rice University is licensed under a Creative Commons Attribution 4.0 International License.