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2.E: Graphs (Optional Exercises)

  • Page ID
    11830
  • 2.2: Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

    Q 2.2.1

    Student grades on a chemistry exam were: 77, 78, 76, 81, 86, 51, 79, 82, 84, 99

    1. Construct a stem-and-leaf plot of the data.
    2. Are there any potential outliers? If so, which scores are they? Why do you consider them outliers?

    Q 2.2.2

    Table contains the 2010 obesity rates in U.S. states and Washington, DC.

    State Percent (%) State Percent (%) State Percent (%)
    Alabama 32.2 Kentucky 31.3 North Dakota 27.2
    Alaska 24.5 Louisiana 31.0 Ohio 29.2
    Arizona 24.3 Maine 26.8 Oklahoma 30.4
    Arkansas 30.1 Maryland 27.1 Oregon 26.8
    California 24.0 Massachusetts 23.0 Pennsylvania 28.6
    Colorado 21.0 Michigan 30.9 Rhode Island 25.5
    Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
    Delaware 28.0 Mississippi 34.0 South Dakota 27.3
    Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
    Florida 26.6 Montana 23.0 Texas 31.0
    Georgia 29.6 Nebraska 26.9 Utah 22.5
    Hawaii 22.7 Nevada 22.4 Vermont 23.2
    Idaho 26.5 New Hampshire 25.0 Virginia 26.0
    Illinois 28.2 New Jersey 23.8 Washington 25.5
    Indiana 29.6 New Mexico 25.1 West Virginia 32.5
    Iowa 28.4 New York 23.9 Wisconsin 26.3
    Kansas 29.4 North Carolina 27.8 Wyoming 25.1
    1. Use a random number generator to randomly pick eight states. Construct a bar graph of the obesity rates of those eight states.
    2. Construct a bar graph for all the states beginning with the letter "A."
    3. Construct a bar graph for all the states beginning with the letter "M."

    S 2.2.2

    1. Example solution for using the random number generator for the TI-84+ to generate a simple random sample of 8 states. Instructions are as follows.
      • Number the entries in the table 1–51 (Includes Washington, DC; Numbered vertically)
      • Press MATH
      • Arrow over to PRB
      • Press 5:randInt(
      • Enter 51,1,8)

      Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

      Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

    A bar graph showing 8 states on the x-axis and corresponding obesity rates on the y-axis.
    Figure 2.2.1 (a)
    This is a bar graph that matches the supplied data. The x-axis shows states, and the y-axis shows percentages.
    Figure 2.2.2 (b)
    This is a bar graph that matches the supplied data. The x-axis shows states, and the y-axis shows percentages.
    Figure 2.2.2 (c)

    For each of the following data sets, create a stem plot and identify any outliers.

    Exercise 2.2.7

    The miles per gallon rating for 30 cars are shown below (lowest to highest).

    19, 19, 19, 20, 21, 21, 25, 25, 25, 26, 26, 28, 29, 31, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 38, 38, 38, 41, 43, 43

    Answer

    Stem Leaf
    1 9 9 9
    2 0 1 1 5 5 5 6 6 8 9
    3 1 1 2 2 3 4 5 6 7 7 8 8 8 8
    4 1 3 3

    The height in feet of 25 trees is shown below (lowest to highest).

    25, 27, 33, 34, 34, 34, 35, 37, 37, 38, 39, 39, 39, 40, 41, 45, 46, 47, 49, 50, 50, 53, 53, 54, 54

    The data are the prices of different laptops at an electronics store. Round each value to the nearest ten.

    249, 249, 260, 265, 265, 280, 299, 299, 309, 319, 325, 326, 350, 350, 350, 365, 369, 389, 409, 459, 489, 559, 569, 570, 610

    Answer

    Stem Leaf
    2 5 5 6 7 7 8
    3 0 0 1 2 3 3 5 5 5 7 7 9
    4 1 6 9
    5 6 7 7
    6 1

    The data are daily high temperatures in a town for one month.

    61, 61, 62, 64, 66, 67, 67, 67, 68, 69, 70, 70, 70, 71, 71, 72, 74, 74, 74, 75, 75, 75, 76, 76, 77, 78, 78, 79, 79, 95

    For the next three exercises, use the data to construct a line graph.

    Exercise 2.2.8

    In a survey, 40 people were asked how many times they visited a store before making a major purchase. The results are shown in Table.

    Number of times in store Frequency
    1 4
    2 10
    3 16
    4 6
    5 4

    Answer

    This is a line graph that matches the supplied data. The x-axis shows the number of times people reported visiting a store before making a major purchase, and the y-axis shows the frequency.
    Figure 2.2.7

    Exercise 2.2.9

    In a survey, several people were asked how many years it has been since they purchased a mattress. The results are shown in Table.

    Years since last purchase Frequency
    0 2
    1 8
    2 13
    3 22
    4 16
    5 9

    Exercise 2.2.10

    Several children were asked how many TV shows they watch each day. The results of the survey are shown in Table.

    Number of TV Shows Frequency
    0 12
    1 18
    2 36
    3 7
    4 2

    Answer

    This is a line graph that matches the supplied data. The x-axis shows the number of TV shows a kid watches each day, and the y-axis shows the frequency.
    Figure 2.2.8

    Exercise 2.2.11

    The students in Ms. Ramirez’s math class have birthdays in each of the four seasons. Table shows the four seasons, the number of students who have birthdays in each season, and the percentage (%) of students in each group. Construct a bar graph showing the number of students.

    Seasons Number of students Proportion of population
    Spring 8 24%
    Summer 9 26%
    Autumn 11 32%
    Winter 6 18%

    Using the data from Mrs. Ramirez’s math class supplied in Exercise, construct a bar graph showing the percentages.

    Answer

    This is a bar graph that matches the supplied data. The x-axis shows the seasons of the year, and the y-axis shows the proportion of birthdays.
    Figure 2.2.9

    Exercise 2.2.12

    David County has six high schools. Each school sent students to participate in a county-wide science competition. Table shows the percentage breakdown of competitors from each school, and the percentage of the entire student population of the county that goes to each school. Construct a bar graph that shows the population percentage of competitors from each school.

    High School Science competition population Overall student population
    Alabaster 28.9% 8.6%
    Concordia 7.6% 23.2%
    Genoa 12.1% 15.0%
    Mocksville 18.5% 14.3%
    Tynneson 24.2% 10.1%
    West End 8.7% 28.8%

    Use the data from the David County science competition supplied in Exercise. Construct a bar graph that shows the county-wide population percentage of students at each school.

    Answer

    This is a bar graph that matches the supplied data. The x-axis shows the county high schools, and the y-axis shows the proportion of county students.
    Figure 2.2.10

    2.3: Histograms, Frequency, Polygons, and Time Series Graphs

    Q 2.3.1

    Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month. Each publisher conducted a survey. In the survey, adult consumers were asked the number of fiction paperbacks they had purchased the previous month. The results are as follows:

    Publisher A
    # of books Freq. Rel. Freq.
    0 10
    1 12
    2 16
    3 12
    4 8
    5 6
    6 2
    8 2
    Publisher B
    # of books Freq. Rel. Freq.
    0 18
    1 24
    2 24
    3 22
    4 15
    5 10
    7 5
    9 1
    Publisher C
    # of books Freq. Rel. Freq.
    0–1 20
    2–3 35
    4–5 12
    6–7 2
    8–9 1
    1. Find the relative frequencies for each survey. Write them in the charts.
    2. Using either a graphing calculator, computer, or by hand, use the frequency column to construct a histogram for each publisher's survey. For Publishers A and B, make bar widths of one. For Publisher C, make bar widths of two.
    3. In complete sentences, give two reasons why the graphs for Publishers A and B are not identical.
    4. Would you have expected the graph for Publisher C to look like the other two graphs? Why or why not?
    5. Make new histograms for Publisher A and Publisher B. This time, make bar widths of two.
    6. Now, compare the graph for Publisher C to the new graphs for Publishers A and B. Are the graphs more similar or more different? Explain your answer.

    Q 2.3.2

    Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all onboard transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera. Following is a summary of the bills for each group.

    Singles
    Amount($) Frequency Rel. Frequency
    51–100 5
    101–150 10
    151–200 15
    201–250 15
    251–300 10
    301–350 5
    Couples
    Amount($) Frequency Rel. Frequency
    100–150 5
    201–250 5
    251–300 5
    301–350 5
    351–400 10
    401–450 10
    451–500 10
    501–550 10
    551–600 5
    601–650 5
    1. Fill in the relative frequency for each group.
    2. Construct a histogram for the singles group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
    3. Construct a histogram for the couples group. Scale the x-axis by $50 widths. Use relative frequency on the y-axis.
    4. Compare the two graphs:
      1. List two similarities between the graphs.
      2. List two differences between the graphs.
      3. Overall, are the graphs more similar or different?
    5. Construct a new graph for the couples by hand. Since each couple is paying for two individuals, instead of scaling the x-axis by $50, scale it by $100. Use relative frequency on the y-axis.
    6. Compare the graph for the singles with the new graph for the couples:
      1. List two similarities between the graphs.
      2. Overall, are the graphs more similar or different?
    7. How did scaling the couples graph differently change the way you compared it to the singles graph?
    8. Based on the graphs, do you think that individuals spend the same amount, more or less, as singles as they do person by person as a couple? Explain why in one or two complete sentences.

    S 2.3.2

    Singles
    Amount($) Frequency Relative Frequency
    51–100 5 0.08
    101–150 10 0.17
    151–200 15 0.25
    201–250 15 0.25
    251–300 10 0.17
    301–350 5 0.08
    Couples
    Amount($) Frequency Relative Frequency
    100–150 5 0.07
    201–250 5 0.07
    251–300 5 0.07
    301–350 5 0.07
    351–400 10 0.14
    401–450 10 0.14
    451–500 10 0.14
    501–550 10 0.14
    551–600 5 0.07
    601–650 5 0.07
    1. See Table and Table.
    2. In the following histogram data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where both boundary values are included).
      This is a histogram that matches the supplied data supplied for singles. The x-axis shows the total charges in intervals of 50 from 50 to 350, and the y-axis shows the relative frequency in increments of 0.05 from 0 to 0.3.
      Figure 2.3.2.1.
    3. In the following histogram, the data values that fall on the right boundary are counted in the class interval, while values that fall on the left boundary are not counted (with the exception of the first interval where values on both boundaries are included).
      This is a histogram that matches the supplied data for couples. The x-axis shows the total charges in intervals of 50 from 100 to 650, and the y-axis shows the relative frequency in increments of 0.02 from 0 to 0.16.
      Figure 2.3.2.2.
    4. Compare the two graphs:
      1. Answers may vary. Possible answers include:
        • Both graphs have a single peak.
        • Both graphs use class intervals with width equal to $50.
      2. Answers may vary. Possible answers include:
        • The couples graph has a class interval with no values.
        • It takes almost twice as many class intervals to display the data for couples.
      3. Answers may vary. Possible answers include: The graphs are more similar than different because the overall patterns for the graphs are the same.
    5. Check student's solution.
    6. Compare the graph for the Singles with the new graph for the Couples:
        • Both graphs have a single peak.
        • Both graphs display 6 class intervals.
        • Both graphs show the same general pattern.
      1. Answers may vary. Possible answers include: Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
    7. Answers may vary. Possible answers include: You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the Couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
    8. Answers may vary. Possible answers include: Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

    Q 2.3.3

    Twenty-five randomly selected students were asked the number of movies they watched the previous week. The results are as follows.

    # of movies Frequency Relative Frequency Cumulative Relative Frequency
    0 5
    1 9
    2 6
    3 4
    4 1
    1. Construct a histogram of the data.
    2. Complete the columns of the chart.

    Use the following information to answer the next two exercises: Suppose one hundred eleven people who shopped in a special t-shirt store were asked the number of t-shirts they own costing more than $19 each.

    A histogram showing the results of a survey.  Of 111 respondents, 5 own 1 t-shirt costing more than $19, 17 own 2, 23 own 3, 39 own 4, 25 own 5, 2 own 6, and no respondents own 7.
    Figure 2.3.3.1.

    Q 2.3.4

    The percentage of people who own at most three t-shirts costing more than $19 each is approximately:

    1. 21
    2. 59
    3. 41
    4. Cannot be determined

    S 2.3.4

    c

    Q 2.3.5

    If the data were collected by asking the first 111 people who entered the store, then the type of sampling is:

    1. cluster
    2. simple random
    3. stratified
    4. convenience

    Q 2.3.6

    Following are the 2010 obesity rates by U.S. states and Washington, DC.

    State Percent (%) State Percent (%) State Percent (%)
    Alabama 32.2 Kentucky 31.3 North Dakota 27.2
    Alaska 24.5 Louisiana 31.0 Ohio 29.2
    Arizona 24.3 Maine 26.8 Oklahoma 30.4
    Arkansas 30.1 Maryland 27.1 Oregon 26.8
    California 24.0 Massachusetts 23.0 Pennsylvania 28.6
    Colorado 21.0 Michigan 30.9 Rhode Island 25.5
    Connecticut 22.5 Minnesota 24.8 South Carolina 31.5
    Delaware 28.0 Mississippi 34.0 South Dakota 27.3
    Washington, DC 22.2 Missouri 30.5 Tennessee 30.8
    Florida 26.6 Montana 23.0 Texas 31.0
    Georgia 29.6 Nebraska 26.9 Utah 22.5
    Hawaii 22.7 Nevada 22.4 Vermont 23.2
    Idaho 26.5 New Hampshire 25.0 Virginia 26.0
    Illinois 28.2 New Jersey 23.8 Washington 25.5
    Indiana 29.6 New Mexico 25.1 West Virginia 32.5
    Iowa 28.4 New York 23.9 Wisconsin 26.3
    Kansas 29.4 North Carolina 27.8 Wyoming 25.1

    Construct a bar graph of obesity rates of your state and the four states closest to your state. Hint: Label the \(x\)-axis with the states.

    S 2.3.7

    Answers will vary.

    Exercise 2.3.6

    Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars. Complete the table.

    Data Value (# cars) Frequency Relative Frequency Cumulative Relative Frequency

    Exercise 2.3.7

    What does the frequency column in Table sum to? Why?

    Answer

    65

    Exercise 2.3.8

    What does the relative frequency column in Table sum to? Why?

    Exercise 2.3.9

    What is the difference between relative frequency and frequency for each data value in Table?

    Answer

    The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

    Exercise 2.3.10

    What is the difference between cumulative relative frequency and relative frequency for each data value?

    Exercise 2.3.11

    To construct the histogram for the data in Table, determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram. Label the horizontal and vertical axes with words. Include numerical scaling.

    An empty graph template for use with this question.
    Figure 2.3.9.

    Answer

    Answers will vary. One possible histogram is shown:

    Figure 2.3.10.

    Exercise 2.3.12

    Construct a frequency polygon for the following:

    1. Pulse Rates for Women Frequency
      60–69 12
      70–79 14
      80–89 11
      90–99 1
      100–109 1
      110–119 0
      120–129 1
    2. Actual Speed in a 30 MPH Zone Frequency
      42–45 25
      46–49 14
      50–53 7
      54–57 3
      58–61 1
    3. Tar (mg) in Nonfiltered Cigarettes Frequency
      10–13 1
      14–17 0
      18–21 15
      22–25 7
      26–29 2

    Exercise 2.3.13

    Construct a frequency polygon from the frequency distribution for the 50 highest ranked countries for depth of hunger.

    Depth of Hunger Frequency
    230–259 21
    260–289 13
    290–319 5
    320–349 7
    350–379 1
    380–409 1
    410–439 1

    Answer

    Find the midpoint for each class. These will be graphed on the x-axis. The frequency values will be graphed on the y-axis values.

    This is a frequency polygon that matches the supplied data. The x-axis shows the depth of hunger, and the y-axis shows the frequency.
    Figure 2.3.11.

    Exercise 2.3.14

    Use the two frequency tables to compare the life expectancy of men and women from 20 randomly selected countries. Include an overlayed frequency polygon and discuss the shapes of the distributions, the center, the spread, and any outliers. What can we conclude about the life expectancy of women compared to men?

    Life Expectancy at Birth – Women Frequency
    49–55 3
    56–62 3
    63–69 1
    70–76 3
    77–83 8
    84–90 2
    Life Expectancy at Birth – Men Frequency
    49–55 3
    56–62 3
    63–69 1
    70–76 1
    77–83 7
    84–90 5

    Exercise 2.3.15

    Construct a times series graph for (a) the number of male births, (b) the number of female births, and (c) the total number of births.

    Sex/Year 1855 1856 1857 1858 1859 1860 1861
    Female 45,545 49,582 50,257 50,324 51,915 51,220 52,403
    Male 47,804 52,239 53,158 53,694 54,628 54,409 54,606
    Total 93,349 101,821 103,415 104,018 106,543 105,629 107,009
    Sex/Year 1862 1863 1864 1865 1866 1867 1868 1869
    Female 51,812 53,115 54,959 54,850 55,307 55,527 56,292 55,033
    Male 55,257 56,226 57,374 58,220 58,360 58,517 59,222 58,321
    Total 107,069 109,341 112,333 113,070 113,667 114,044 115,514 113,354
    Sex/Year 1871 1870 1872 1871 1872 1827 1874 1875
    Female 56,099 56,431 57,472 56,099 57,472 58,233 60,109 60,146
    Male 60,029 58,959 61,293 60,029 61,293 61,467 63,602 63,432
    Total 116,128 115,390 118,765 116,128 118,765 119,700 123,711 123,578

    Answer

    Figure 2.3.12.

    Exercise 2.3.16

    The following data sets list full time police per 100,000 citizens along with homicides per 100,000 citizens for the city of Detroit, Michigan during the period from 1961 to 1973.

    Year 1961 1962 1963 1964 1965 1966 1967
    Police 260.35 269.8 272.04 272.96 272.51 261.34 268.89
    Homicides 8.6 8.9 8.52 8.89 13.07 14.57 21.36
    Year 1968 1969 1970 1971 1972 1973
    Police 295.99 319.87 341.43 356.59 376.69 390.19
    Homicides 28.03 31.49 37.39 46.26 47.24 52.33
    1. Construct a double time series graph using a common x-axis for both sets of data.
    2. Which variable increased the fastest? Explain.
    3. Did Detroit’s increase in police officers have an impact on the murder rate? Explain.
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