Unit 1 Describing Data

Review

  1. Determine whether the data are qualitative or quantitative:
    1. the colors of automobiles on a used car lot qualitative
    2. the numbers on the shirts of a soccer team qualitative
    3. the number of seats in a movie theater quantitative
    4. a list of house numbers on your street qualitative
    5. the ages of a sample of 350 employees of a large hospital quantitative
  2. Identify the population and the sample.
    1. A survey will be given to 100 students randomly selected from the freshmen class at Lincoln High School.

      Sample: 100 freshmen students at Lincoln High School

      Polulation: freshman students at Lincoln High School

    2. Fifty bottles of water were randomly selected from a large collection of bottles in a company's warehouse.

      Sample: 50 bottles in the warehouse

      Polulation: bottles in the warehouse

    3. You’re interested in knowing what percent of all households in a large city have a single woman as the head of the household. To estimate this percentage, you conduct a survey with 200 households and determine how many of these 200 are headed by a single woman.

      Sample: 200 households surveyed in the city

      Polulation: households in the city

  3. Here we list the 20 countries that emitted the most carbon dioxide in 2015.
    Rank and Country 2015 Per Capita Carbon Dioxide Emissions from Fuel Combustion (metric tons)
    1 China 6.6
    2 United States 15.5
    3 India 1.6
    4 Russia 10.2
    5 Japan 9.0
    6 Germany 8.9
    7 South Korea 11.6
    8 Iran 7.0
    9 Canada 15.3
    10 Saudia Arabia 16.9
    11 Brazil 2.2
    12 Mexico 3.7
    13 Indonesia 1.7
    14 south Africa 7.8
    15 United Kingdom 6.0
    16 Australia 15.8
    17 Italy 5.5
    18 Turkey 4.1
    19 France 4.4
    20 Poland 7.3

    Construct the following using the data: Frequency Distribution, Relative Frequency Distribution, Cumulative Frequency Distribution, Histogram, Dot Plot, Stem and Leaf. For the frequency distribution use 6 classes and start the first class at 0.

    Frequency
    EMISSIONS FREQUENCY

    0 – 2.9

    3

    3 - 5.9

    4

    6 - 8.9

    6

    9 - 11.9

    3

    12 - 14.9

    0

    15 - 17.9

    4
    Relative Frequency
    EMISSIONS RELATIVE
    FREQUENCY

    0 - 2.9

    15%

    3 – 5.9

    20%

    6 - 8.9

    30%

    9 - 11.9

    15%

    12 - 14.9

    0%

    15 - 17.9

    20%
    Cumulative Frequency
    EMISSIONS CUMULATIVE
    FREQUENCY

    0 - 2.9

    3

    3 - 5.9

    7

    6 - 8.9

    13

    9 - 11.9

    16

    12 - 14.9

    16

    15 - 17.9

    20
    Stem Leaves

    1

    6  7

    2

    2

    3

    7

    4

    1  4

    5

    6

    0  6

    7

    0  3  8

    8

    9

    9

    0

    10

    2

    11

    6

    12

    13

    14

    15

    3  5  8

    16

    9

    Legend 16

    9 = 16.9

    A histogram representing the 2015 Per Capita Carbon Dioxide Emissions from Fuel Combustion for the Top 20 Polluting Countries. The horizontal axis represents the classes of per capita emissions in metric tons. The classes are 0-2.9, 3-5.9, 6-8.9, 9-11.9, 12-14.9, and 15-17.9. The vertical axis represents the number of countries in each class and goes from 0 to 7 counting by 1. The number in each class is 3,4,6,3,0, and 4 respectively.

    A dot plot representing the 2015 Per Capita Carbon Dioxide Emissions from Fuel Combustion for the Top 20 Polluting Countries. The horizontal axis represents the per capita emissions in metric tons. It starts at 0 and ends at 18, counting by 1. The horizontal axis represents the number of countries and goes from 0 to 4, counting by 1. There are 3 dots at 2 metric tons, 3 dots at 4 metric tons, 2 dots at 6 metric tons, 3 dots at 7 metric tons, 1 dot at 8 metric tons, 2 dots at 9 metric tons, 1 dot at 12 metric tons, 1 dot at 15 metric tons, 2 dots at 16 metric tons, and 1 dot at 17 metric tons.




  4. Below is a random sample of life expectancies from 20 countries:
    70.5 65 70 51.5 57.5 61 78.5 61 72 64.5
    56.5 73 69 52.5 78.5 54 74.5 76 70 68.5
    1. Make a frequency table of the life expectancies.

      Use 6 classes and start the first class at 50.

      Class Frequency

      50.0 – 54.9

      3

      55.0 – 59.9

      2

      60.0 – 64.9

      3

      65.0 – 69.9

      3

      70.0 – 74.9

      6

      75.0 – 79.9

      3

    2. Answer the following questions based on your histogram:
      1. What are the class midpoints?

        52.45, 57.45, 62.45, 67.45, 72.45, 77.45

      2. What are your lower class limits?

        50.0, 55.0, 60.0, 65.0, 70.0, 75.0

      3. What are your upper class limits?

        54.9, 59.9, 64.9, 69.9, 74.9, 79.9

      4. Draw a histogram:

        A histogram representing the life expectancies of people in 20 countries. The horizontal axis is the midpoint of each class: 52.45, 57.45, 62.45, 67.45, 72.45, and 77.45. The vertical axis is numbered from 0 to 8 counting by 2. The first bar is at 3, the second one at 2, the third one at 3, the fourth one at 3, the fifth one at 6 and the last one at 3.

      5. Use the same data to create a relative frequency distribution:
        Classes Relative Frequency

        50.0 - 54.9

        3/20 = 15%

        55.0 - 59.9

        2/20 = 10%

        60.0 - 64.9

        3/20 = 15%

        65.0 - 69.9

        3/20 = 15%

        70.0 - 74.9

        6/20 = 30%

        75.0 - 79.9

        3/20 = 15%

  5. Use the following data to complete a-e:

    AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest):

    3 4 8 8 10 11 12 13 14 15
    15 16 16 17 17 18 21 22 22 24
    24 25 26 26 27 27 29 29 31 32
    33 33 34 34 35 37 40 44 44 47
    1. Calculate the measures of center from the given list of numbers.

      Mean: 23.6

      Median: 24

      Mode: multi-modal

      Midrange: 25

    2. Create a frequency table using 2 as the lower limit of the first class and a class width of 8.
      CLASS FREQUENCY

      2 - 9

      4

      10 - 17

      11

      18 - 25

      7

      26 - 33

      10

      34 - 41

      5

      42 - 49

      3
    3. ESTIMATE the mean of the data using the frequency table. Mean = 23.5
    4. ESTIMATE the median of the data using the frequency table. First, identify the position of the median. Which CLASS in the frequency table contains the median?

      The position of the median is 41/2 = 20.5, so the 21st term, which is in the third CLASS 18 - 25.



  6. These are the volumes (in ounces) of randomly selected cans of Coke. Find the Mean, median, mode, and midrange.
    12.3 12.0 12.1 12.3 12.2 12.3 12.2
    Mean = 12.2 Median = 12.2 Mode = 2.3 Midrange = 2.15

  7. Find the mean of the following frequency distribution:
    GPA FREQUENCY

    CLASS MIDPOINT

    Frequency x Midpoint
    0 - 0.9 4

    0.45

    1.8
    1 - 1.9 7

    1.45

    10.15
    2 - 2.9 12

    2.45

    29.4
    3 - 3.9 15

    3.45

    51.75
    4 - 4.9 6

    4.45

    26.7

    SUM = 44

    SUM = 119.8

    119.8/44 = 2.72

    MEAN = 2.72



  8. What is the shape of the data represented in the frequency distribution?

    Skewed to the left



  9. The ages of the employees at a local newspaper are given. Use the data to complete a-d:
    20 26 52 30 21 36 34 60 57 51 56 63 42
    1. Calculate the measures of variation from the given list of numbers. Range = 43 Variance = 232.6 Standard Deviation = 15.3
    2. Create a frequency table using 20 as the lower limit of the first class and a class width of 10.
      CLASS FREQUENCY

      MIDPOINT

      20 -29

      3

      24.5

      30 - 39

      3

      34.5

      40 - 49

      1

      44.5

      50 - 59

      4

      54.5

      60 - 69

      2

      64.5
    3. ESTIMATE the mean of the ages using the frequency table. Mean = 43.7
    4. ESTIMATE the standard deviation of the ages using the frequency table. Standard Deviation = 15.0

  10. Use the frequency table to estimate the mean and standard deviation of ticketed speeds:
    Speed in mph of Driver
    Ticketed in 30 mph Zone    		 Frequency of Speed
    Reported on the Ticket

    Midpoint
    42 - 45 10

    43.5
    46 - 49 14

    47.5
    50 - 53 7

    51.5
    54 - 57 3

    55.5
    58 - 61 1

    59.5
    1. Estimate the mean of the data: Mean = 48.2
    2. Estimate the standard deviation of the data: Standard Deviation = 4.2

    11. FIVE-NUMBER SUMMARIES AND PERCENTILES

  11. The circumference measurements (in cm) of a sample of randomly selected trees on a farmer’s property is given below. Use the data to answer the following questions.
    18 18 19 24 31 34 37 37 38 39
    40 41 49 51 51 52 53 55 83 112
    1. CALCULATE THE FOLLOWING

      Mean: 44.1

      Median: 39.5

      Mode: 18, 37, 51

      MidRange: 65

      Range: 94

      Variance: 492.8

      Standard Deviation: 22.2

      Q1: 32.5

      Q3: 51.5

      IQR: 51.5 - 32.5 = 19

      Are there any outliers in the data?

      Yes, 83 and 112 are outliers

      Lower Outlier Limit: Q1 – 1.5*IQR   32 – 1.5*19 = 4

      Any data point less than 4 is an outlier.

      Upper Outlier Limit: Q3 + 1.5*IQR   51.5 + 1.5*19 = 80

      Any data point greater than 80 is an outlier.

    2. CREATE A BOXPLOT OF THE DATA: Mark outliers clearly on the boxplot.

      A box and whisker plot with a 5 number summary of 18, 32.5, 39.5, 51.5, and 112. There are 2 outliers on the right side of the graph.

  12. Review of Frequency Distributions
    1. Use the data to construct a Frequency Distribution Table: Begin with a lower class limit of 30 and a class width of 15.

      32   49   53   57   61   64   66   68   68   68   71   72   72   75   79   80   83   85   90   93

      CLASS FREQUENCY RELATIVE FREQUENCY
      30- 44

      1

      0.05

      45-59

      3

      0.15

      60-74

      9

      0.45

      75-89

      5

      0.25

      90-104

      2

      0.10

      Cumulative Class Cumulative Frequency

      Less than 45

      1

      Less than 60

      4

      Less than 75

      13

      Less than 90

      18

      Less than 105

      20
    2. Find the following using the Frequency Table (not the Relative or Cumulative summary information):
      • •Lower Class Limit of the 3rd Class 60
      • Lower Class Boundary of the 3rd Class 59.5
      • Midpoint of the 3rd Class 67
    3. Use the Frequency Distribution (not Relative or Cumulative) to draw a Histogram of the data:

      A frequency distribution (bar graph) with the x-axis labeled from 30 to 105 in intervals of 15. The frequency is labeled on the y-axis from 0 to 8 in intervals of 2. The first bar goes from 30 to 45 with a height of 1. The second bar goes from 45 to 60 with a height of 3. The third bar goes from 60 to 75 with a height of 9. The fourth bar goes from 75 to 90 with a height of 5, and the fifth and final bar goes from 90 to 105 with a height of 2.