12.8 Exercises

1. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.

  1. What are the dependent and independent variables?
  2. Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.
  3. 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?

  1. [latex]y = 6x + 8[/latex]
  2. [latex]y + 7 = 3x[/latex]
  3. [latex]y – x = 8x^2[/latex]
  4. [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)?

Adults and Adolescents only, United States
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].

  1. What are the independent and dependent variables?
  2. 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].

  1. What are the independent and dependent variables?
  2. How many pounds of soil does the shoreline lose in a year?
  3. 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.

  1. What are the slope and y-intercept? Interpret their meaning.
  2. 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.

  1. 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.
  2. A study is done to determine if 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 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?

This is a scatterplot with several points plotted in the first quadrant. The points form a clear pattern, moving upward to the right. The points do not line up , but the overall pattern can be modeled with a line.

10. Does the scatter plot appear linear? Strong, moderate, or weak? Positive or negative?

This is a scatterplot with several points plotted in the first quadrant. The points move downward to the right. The overall pattern can be modeled with a line, but the points are widely scattered.

11. Does the scatter plot appear linear? Strong, moderate, or weak? Positive or negative?

This is a scatter plot with several points plotted all over the first quadrant. There is no pattern.

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
  1. Draw a scatter plot of the data.
  2. Use regression to find the equation for the line of best fit.
  3. Draw the line of best fit on the scatter plot.
  4. What is the slope of the line of best fit? What does it represent?
  5. 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.

  1. What would you predict the sales to be on day 60?
  2. 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.

  1. How many acres will be left to mow after 20 hours of work?
  2. How many acres will be left to mow after 100 hours of work?
  3. 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.

Adults and Adolescents only, United States
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
  1. Graph “year” versus “# AIDS cases diagnosed” (plot the scatter plot). Do not include pre-1981 data.
  2. Calculate the correlation coefficient.
  3. Interpret the correlation coefficient.
  4. Find the linear regression equation.
  5. Interpret the slope of the linear regression equation.
  6. What is the predicted number of diagnosed cases for the year 1985?
  7. What is the predicted number of diagnosed cases for the year 1970?  Why doesn’t this answer make sense?
  8. Calculate the coefficient of determination.
  9. Interpret the coefficient of determination.
  10. Calculate the standard error of the estimate.
  11. 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
  1. 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.
  2. Calculate the least squares (best–fit) line.
  3. Interpret the slope of the least squares line.
  4. Predict the number of deaths people aged 40.
  5. Find the correlation coefficient.
  6. Interpret the correlation coefficient.
  7. Find the coefficient of determination.
  8. Interpret the coefficient of determination.
  9. Find the standard error of the estimate.
  10. 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
  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. Find the correlation coefficient
  4. Interpret the correlation coefficient.
  5. Find the linear regression equation.
  6. Interpret the slope of the linear regression equation.
  7. 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?
  8. What is the estimated life expectancy for someone born in 1982?
  9. 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.
  10. Calculate the coefficient of determination.
  11. Interpret the coefficient of determination.
  12. Calculate the standard error of the estimate.
  13. 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
  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. Find the correlation coefficient
  4. Interpret the correlation coefficient.
  5. Find the linear regression equation.
  6. Interpret the slope of the linear regression equation.
  7. What is the estimated maximum value for restaurants on page 10?
  8. What is the estimated maximum value for restaurants on page 70?
  9. 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.
  10. Calculate the coefficient of determination.
  11. Interpret the coefficient of determination.
  12. Calculate the standard error of the estimate.
  13. 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
    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. Find the correlation coefficient
    4. Interpret the correlation coefficient.
    5. Find the linear regression equation.
    6. Interpret the slope of the linear regression equation.
    7. What is the estimated gold medal time for 1932?
    8. What is the estimated gold medal time for 1984?
    9. Calculate the coefficient of determination.
    10. Interpret the coefficient of determination.
    11. Calculate the standard error of the estimate.
    12. 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
    1. Using “stories” as the independent variable and “height” as the dependent variable, draw a scatter plot of the ordered pairs.
    2. Find the correlation coefficient
    3. Interpret the correlation coefficient.
    4. Find the linear regression equation.
    5. Interpret the slope of the linear regression equation.
    6. What is the estimated height for a 32 story building?
    7. What is the estimated height for a 94 story building?
    8. 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.
    9. Calculate the coefficient of determination.
    10. Interpret the coefficient of determination.
    11. Calculate the standard error of the estimate.
    12. 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
    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. Find the correlation coefficient
    4. Interpret the correlation coefficient.
    5. Find the linear regression equation.
    6. Interpret the slope of the linear regression equation.
    7. Calculate the coefficient of determination.
    8. Interpret the coefficient of determination.
    9. Calculate the standard error of the estimate.
    10. 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
    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. Find the correlation coefficient
    4. Interpret the correlation coefficient.
    5. Find the linear regression equation.
    6. Interpret the slope of the linear regression equation.
    7. What is the estimated heart rate for a swim time of 34.75 minutes?
    8. Calculate the coefficient of determination.
    9. Interpret the coefficient of determination.
    10. Calculate the standard error of the estimate.
    11. 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
    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. Find the correlation coefficient
    4. Interpret the correlation coefficient.
    5. Find the linear regression equation.
    6. Interpret the slope of the linear regression equation.
    7. What is the estimated percent of workers paid hourly rates in 1988?
    8. Calculate the coefficient of determination.
    9. Interpret the coefficient of determination.
    10. Calculate the standard error of the estimate.
    11. 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
    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. Find the correlation coefficient
    4. Interpret the correlation coefficient.
    5. Find the linear regression equation.
    6. Interpret the slope of the linear regression equation.
    7. What is the estimated average height for a one-year old?
    8. 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.
    9. Calculate the coefficient of determination.
    10. Interpret the coefficient of determination.
    11. Calculate the standard error of the estimate.
    12. Interpret the standard error of the estimate.

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