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Math  /  Data & Statistics

QuestionHomework The accompanying data set lists the numbers of children of world leaders. Use the data to construct a frequency distribution using six classes and to create a frequency polygon. Describe any patterns.
Click the icon to view the data set.
Complete the frequency distribution table below. Use the minimum data entry as the limit of the first class.

Studdy Solution

STEP 1

1. The data set lists the numbers of children of world leaders: 6, 3, 0, 5, 14, 15, 3, 2, 5, 2, 11, 16, 1, 2, 0, 0, 7, 7, 1, 3, 3, 4, 10, 6, 12, 1, 5, 6, 4, 11.
2. We need to construct a frequency distribution using six classes.
3. We need to create a frequency polygon from the frequency distribution.
4. The minimum data entry should be used as the limit of the first class.

STEP 2

1. Determine the range and class width.
2. Construct the frequency distribution table with six classes.
3. Calculate the midpoints for each class.
4. Create a frequency polygon using the midpoints and frequencies.
5. Describe any patterns observed.

STEP 3

Identify the minimum and maximum values in the data set.
Minimum value: 00
Maximum value: 1616

STEP 4

Calculate the range of the data.
Range=Maximum value−Minimum value=16−0=16 \text{Range} = \text{Maximum value} - \text{Minimum value} = 16 - 0 = 16

STEP 5

Determine the class width.
Class width=RangeNumber of classes=166≈2.67 \text{Class width} = \frac{\text{Range}}{\text{Number of classes}} = \frac{16}{6} \approx 2.67
Since class width should be a whole number, round up to the next whole number.
Class width=3 \text{Class width} = 3

STEP 6

Construct the classes for the frequency distribution table. Start from the minimum value and add the class width to determine the class limits.
Class 1: 0−20 - 2
Class 2: 3−53 - 5
Class 3: 6−86 - 8
Class 4: 9−119 - 11
Class 5: 12−1412 - 14
Class 6: 15−1715 - 17

STEP 7

Count the number of data points in each class to determine the frequencies.
Class 1: 0−20 - 2: 8 (0, 0, 0, 2, 2, 2, 1, 1)
Class 2: 3−53 - 5: 7 (3, 3, 3, 3, 4, 5, 5)
Class 3: 6−86 - 8: 6 (6, 6, 6, 7, 7, 6)
Class 4: 9−119 - 11: 4 (10, 11, 11, 10)
Class 5: 12−1412 - 14: 2 (12, 14)
Class 6: 15−1715 - 17: 2 (15, 16)

STEP 8

Complete the frequency distribution table.
ClassFrequency0−283−576−869−11412−14215−172\begin{array}{|c|c|} \hline \text{Class} & \text{Frequency} \\ \hline 0 - 2 & 8 \\ 3 - 5 & 7 \\ 6 - 8 & 6 \\ 9 - 11 & 4 \\ 12 - 14 & 2 \\ 15 - 17 & 2 \\ \hline \end{array}

STEP 9

Calculate the midpoints for each class.
Midpoint=Lower limit+Upper limit2\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}
Class 1 midpoint: 0+22=1 \frac{0 + 2}{2} = 1
Class 2 midpoint: 3+52=4 \frac{3 + 5}{2} = 4
Class 3 midpoint: 6+82=7 \frac{6 + 8}{2} = 7
Class 4 midpoint: 9+112=10 \frac{9 + 11}{2} = 10
Class 5 midpoint: 12+142=13 \frac{12 + 14}{2} = 13
Class 6 midpoint: 15+172=16 \frac{15 + 17}{2} = 16

STEP 10

Complete the frequency distribution table with midpoints.
ClassFrequencyMidpoint0−2813−5746−8679−1141012−1421315−17216\begin{array}{|c|c|c|} \hline \text{Class} & \text{Frequency} & \text{Midpoint} \\ \hline 0 - 2 & 8 & 1 \\ 3 - 5 & 7 & 4 \\ 6 - 8 & 6 & 7 \\ 9 - 11 & 4 & 10 \\ 12 - 14 & 2 & 13 \\ 15 - 17 & 2 & 16 \\ \hline \end{array}

STEP 11

Create a frequency polygon using the midpoints and frequencies. Plot the midpoints on the x-axis and the frequencies on the y-axis, and connect the points with straight lines.
Plot the points: (1,8)(1, 8), (4,7)(4, 7), (7,6)(7, 6), (10,4)(10, 4), (13,2)(13, 2), (16,2)(16, 2)
Connect the points with straight lines to form the frequency polygon.

STEP 12

Describe any patterns observed in the frequency distribution and frequency polygon.
The frequency distribution and frequency polygon indicate that the majority of world leaders have between 00 and 22 children. The number of leaders with 33 to 55 children is also relatively high. There is a gradual decrease in frequency as the number of children increases beyond 55. Most leaders have fewer than 99 children, with very few having 1212 or more.

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