Question

i can compare two groups of numeric data.

1. in 5th period, a survey is conducted to determine the number of family members in each students family. the results from the band and choir classes are shown below.

band class
choir class
a. what is the median number of family members in band class? choir class?
b. compare the distribution of data.
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c. what is the mode number of family members in band class? choir class?
d. what is the range number of family members in band class? choir class?
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Explanation:

Step1: Count the number of data points for Band Class

For Band Class: \(2 + 4+6 + 7+3 + 2=24\) data points. Since \(n = 24\) (even), the median is the average of the \(\frac{n}{2}=12^{th}\) and \((\frac{n}{2}+ 1)=13^{th}\) ordered - values.
Counting the dots: \(2\) (for \(2\) family members) \(+4\) (for \(3\) family members) \(+6\) (for \(4\) family members) \(=12\). So the \(12^{th}\) and \(13^{th}\) values are \(4\). The median for Band Class is \(4\).

Step2: Count the number of data points for Choir Class

For Choir Class: \(1+4 + 6+7+6 + 3=27\) data points. Since \(n = 27\) (odd), the median is the \(\frac{n + 1}{2}=14^{th}\) ordered - value.
Counting the dots: \(1\) (for \(2\) family members) \(+4\) (for \(3\) family members) \(+6\) (for \(4\) family members) \(=11\). Then we need \(3\) more values from the \(5\) family - member group. So the \(14^{th}\) value is \(5\).

Step3: Find the mode for Band Class

The mode is the value with the highest frequency. In the Band Class dot - plot, the number of family members with the most dots is \(5\) (7 dots).

Step4: Find the mode for Choir Class

In the Choir Class dot - plot, the number of family members with the most dots is \(5\) (7 dots).

Step5: Calculate the range for Band Class

The range is \(R=\text{Max}-\text{Min}\). For Band Class, \(\text{Max}=7\) and \(\text{Min}=2\), so \(R = 7-2=5\).

Step6: Calculate the range for Choir Class

For Choir Class, \(\text{Max}=7\) and \(\text{Min}=2\), so \(R = 7 - 2=5\).

Answer:

a. Median of Band Class: \(4\), Median of Choir Class: \(5\)
b. Band Class: Asymmetric distribution (skewed to the left or right depending on the tail). Choir Class: Symmetric distribution (approximately bell - shaped around \(4\) and \(5\)).
c. Mode of Band Class: \(5\), Mode of Choir Class: \(5\)
d. Range of Band Class: \(5\), Range of Choir Class: \(5\)