Answer:
[tex]Mean = 100[/tex]
[tex]Median = 96[/tex]
Step-by-step explanation:
Given
[tex]C_v = 30\%[/tex] --- coefficient of variation
[tex]mode = 88[/tex]
[tex]Skp = 0.4[/tex]
Required
The mean and the median
The coefficient of variation is calculated using:
[tex]C_v = \frac{\sigma}{\mu}[/tex]
Where:
[tex]\mu \to[/tex] mean
So:
[tex]30\% = \frac{\sigma}{\mu}[/tex]
Express percentage as decimal
[tex]0.30 = \frac{\sigma}{\mu}[/tex]
Make [tex]\sigma[/tex] the subject
[tex]\sigma = 0.30\mu[/tex]
The coefficient of skewness is calculated using:
[tex]Skp = \frac{\mu - Mode}{\sigma}[/tex]
This gives:
[tex]0.4 = \frac{\mu - 88}{\sigma}[/tex]
Make [tex]\sigma[/tex] the subject
[tex]\sigma = \frac{\mu - 88}{0.4 }[/tex]
Equate both expressions for [tex]\sigma[/tex]
[tex]0.30\mu = \frac{\mu - 88}{0.4 }[/tex]
Cross multiply
[tex]0.4*0.30\mu = \mu - 88[/tex]
[tex]0.12\mu = \mu - 88[/tex]
Collect like terms
[tex]0.12\mu - \mu = - 88[/tex]
[tex]-0.88\mu = - 88[/tex]
Divide both sides by -0.88
[tex]\mu = 100[/tex]
Hence:
[tex]Mean = 100[/tex]
Calculate [tex]\sigma[/tex]
[tex]\sigma = 0.30\mu[/tex]
[tex]\sigma = 0.30 * 100[/tex]
[tex]\sigma = 30[/tex]
So:
Also, the coefficient of skewness is calculated using:
[tex]Skp = \frac{3 * (Mean - Median)}{\sigma}[/tex]
[tex]0.4= \frac{3 * (100 - Median)}{30}[/tex]
Multiply both sides by 30
[tex]0.4*30= 3 * (100 - Median)[/tex]
Divide both sides by 3
[tex]0.4*10= 100 - Median[/tex]
[tex]4= 100 - Median[/tex]
Collect like terms
[tex]Median = 100 - 4[/tex]
[tex]Median = 96[/tex]
