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Math  /  Data & Statistics

QuestionQuestion 6, 2.1.36 Part 1 of 3 HW Score: 81.39%,6.5181.39 \%, 6.51 of 8 points Points: 0.96 of 1 Sar 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 lower limit of the first class.

Studdy Solution

STEP 1

1. The data set consists of the number of children of world leaders.
2. The minimum data entry is used as the lower limit of the first class.
3. The total number of classes to be used is six.
4. The frequency distribution table will include columns for Class, Frequency, and Midpoint.
5. The frequency polygon will be plotted using the midpoints and frequencies of the classes.

STEP 2

1. Calculate the range of the data.
2. Determine the class width.
3. Construct the class intervals.
4. Tally the data into the frequency distribution table.
5. Calculate the midpoints for each class.
6. Plot the frequency polygon using the midpoints and frequencies.
7. Describe any patterns observed in the frequency polygon.

STEP 3

Identify the minimum and maximum values in the data set.
Minimum value: 0 Maximum value: 16

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. Since we need six classes, divide the range by the number of classes and round up to the next whole number if necessary.
Class width=⌈RangeNumber of classes⌉=⌈166⌉≈3\text{Class width} = \left\lceil \frac{\text{Range}}{\text{Number of classes}} \right\rceil = \left\lceil \frac{16}{6} \right\rceil \approx 3

STEP 6

Construct the class intervals, starting from the minimum value.
Class 1: 0−2Class 2: 3−5Class 3: 6−8Class 4: 9−11Class 5: 12−14Class 6: 15−17\begin{array}{c} \text{Class 1: } 0-2 \\ \text{Class 2: } 3-5 \\ \text{Class 3: } 6-8 \\ \text{Class 4: } 9-11 \\ \text{Class 5: } 12-14 \\ \text{Class 6: } 15-17 \\ \end{array}

STEP 7

Count the frequency of data entries within each class interval and tally them in the frequency distribution table.
ClassFrequency0−293−586−859−11312−14115−174\begin{array}{|c|c|} \hline \text{Class} & \text{Frequency} \\ \hline 0-2 & 9 \\ 3-5 & 8 \\ 6-8 & 5 \\ 9-11 & 3 \\ 12-14 & 1 \\ 15-17 & 4 \\ \hline \end{array}

STEP 8

Calculate the midpoints for each class interval using the formula Midpoint=Lower limit+Upper limit2\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}.
ClassMidpoint0−20+22=13−53+52=46−86+82=79−119+112=1012−1412+142=1315−1715+172=16\begin{array}{|c|c|} \hline \text{Class} & \text{Midpoint} \\ \hline 0-2 & \frac{0+2}{2} = 1 \\ 3-5 & \frac{3+5}{2} = 4 \\ 6-8 & \frac{6+8}{2} = 7 \\ 9-11 & \frac{9+11}{2} = 10 \\ 12-14 & \frac{12+14}{2} = 13 \\ 15-17 & \frac{15+17}{2} = 16 \\ \hline \end{array}

STEP 9

Plot the frequency polygon using the midpoints and frequencies from the table.
MidpointFrequency194875103131164\begin{array}{|c|c|} \hline \text{Midpoint} & \text{Frequency} \\ \hline 1 & 9 \\ 4 & 8 \\ 7 & 5 \\ 10 & 3 \\ 13 & 1 \\ 16 & 4 \\ \hline \end{array}
Then, plot these points on a graph and connect them with straight lines.

STEP 10

Describe any patterns observed in the frequency polygon.
The frequency polygon shows a peak at the midpoint of 1, indicating that most world leaders have between 0 and 2 children. The frequency decreases as the number of children increases, with a slight increase again at the midpoint of 16, showing that a few world leaders have a significantly higher number of children.

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