Question
The data in this question will be utilized for the next three questions. The number of pets a group of students has, are given below.
Number of pets: 0-2 2-4 4-6 6-8 8-10 10-12 12-14
Number of students: 1 2 1 5 6 2 3
Calculate
(1) Mean (2) Sample Variance (3) Sample Standard Deviation
Answer:
[tex](1)[/tex] [tex]\bar x = 8.1[/tex]
[tex](2)[/tex] [tex]\sigma^2 = 6.341[/tex]
[tex](3)[/tex] [tex]\sigma = 2.518[/tex]
Step-by-step explanation:
Given
The above data
First, we calculate the class midpoint (x)
Pets: [tex]0-2[/tex] [tex]2-4[/tex] [tex]4-6[/tex] [tex]6-8[/tex] [tex]8-10[/tex] [tex]10-12[/tex] [tex]12-14[/tex]
x 1 3 5 7 9 11 13
f: 1 2 1 5 6 2 3
The class midpoint (x) is calculated by the average of each group.
For pets: 0 - 2.
[tex]x = \frac{0 + 2}{2}= \frac{2}{2} = 1[/tex]
The same is done for other groups.
Solving (a): Mean
[tex]\bar x = \frac{\sum fx}{\sum f}[/tex]
This gives:
[tex]\bar x = \frac{1 * 1 + 3 * 2 + 5 * 1 + 7 * 5 + 9 * 6 + 11 * 2 + 13 * 3}{1 + 2 + 1 + 5 + 6 + 2 + 3}[/tex]
[tex]\bar x = \frac{162}{20}[/tex]
[tex]\bar x = 8.1[/tex]
Solving (b): Sample Variance
This is calculated as:
[tex]\sigma^2 = \frac{\sum(x_i - \bar x)^2}{\sum f - 1}[/tex]
So:
[tex]\sigma^2 = \frac{(1 - 8.1)^2+(3 - 8.1)^2+(5 - 8.1)^2+(7 - 8.1)^2+(9 - 8.1)^2+(11 - 8.1)^2+(13 - 8.1)^2}{20 - 1}[/tex]
[tex]\sigma^2 = \frac{120.47}{19}[/tex]
[tex]\sigma^2 = 6.341[/tex]
Solving (c): The sample standard deviation
This is calculated as:
[tex]\sigma = \sqrt{\sigma^2[/tex]
[tex]\sigma = \sqrt{6.341[/tex]
[tex]\sigma = 2.518[/tex]
