Given the table:
Number of pairs of shoes 4 5 6 7 8 9 10
Frequency 6 11 11 9 8 9 7
Let's solve for the following:
• (a). The mean
To find the mean, we have:
[tex]mean=\frac{(4*6)+(11*5)+(11*6)+(9*7)+(8*8)+(9*9)+(10*7)}{61}[/tex]Solving further:
[tex]\begin{gathered} mean=\frac{423}{61} \\ \\ mean=6.9 \end{gathered}[/tex]The mean is 6.9
• (b). The median
The median is the middle value.
To find the median, we have:
[tex]\frac{n+1}{2}=\frac{61+1}{2}=\frac{62}{2}=31[/tex]The median will be the 31st number.
From the table, the 31st number is = 7
Therefore, the median is 7
• (C). The sample standard deviation
To find the standard deviation, apply the formula:
[tex]s=\sqrt{\frac{\Sigma f(x-x^{\prime})^2}{n-1}}[/tex]Thus, we have:
[tex]\begin{gathered} s=\sqrt{\frac{6(4-6.9)^2+11(5-6.9)^2+11(6-6.9)^2+9(7-6.9)^2+8(8-6.9)^2+9(9-6.9)^2+7(10-6.9)^2}{61-1}} \\ \\ s=\sqrt{3.59683} \\ \\ s=1.9 \end{gathered}[/tex]The standard deviation is 1.9
• (d). The first quartile.
The first quartile is the median of the lower half of the data.
To lower half of data is from 1 to 30th data.
The median of the lower half will be:
[tex]\frac{30}{2}=15[/tex]This means the value in the 15th frequency is the lower quartile.
The value in the 15th frequency is 5.
Therefore, the first quartile is 5.
• (e). Upper quartile
This is the median of the upper half of the data.
The median is the frequency from 32 to 61 frequency.
The median will be:
[tex]32+15=47[/tex]The 47th data.
The value of the 47th data is 9
Therefore, the third quartile is 9.
• (f). What percent have at least 5 pairs of shoes?
Only 6 respondents have less than 5 pairs of shoes.
To find the percent with at least 5 pairs of shoes, we have:
[tex]\begin{gathered} \frac{61-5}{61}*100 \\ \\ =\frac{56}{61}*100 \\ \\ =0.9180*100 \\ \\ =91.8\text{ \%} \end{gathered}[/tex]• (g). 28% have fewer than how many pairs of shoes?
28% have fewer than 6 pairs of shoes.
ANSWER:
• Mean = 6.9
,• Median = 7
,• Standard deviation = 1.9
,• First quartile = 5
,• Third quartile = 9
,• At least 5 pairs = 91.8%
,• 28% have fewer than, ,6 pairs of shoes.