Sure, let's solve for the variance step-by-step.
1. Understand the Definitions and Given Values:
- Coefficient of Variation (CV): Measure of relative variability, defined as the ratio of the standard deviation to the mean, expressed as a percentage.
- Given Values:
- Coefficient of Variation (CV): [tex]\( 9\% \)[/tex] or [tex]\( 0.09 \)[/tex] (since percentages are converted to decimal by dividing by 100).
- Mean ([tex]\( \mu \)[/tex]): 40.
2. Formula for Coefficient of Variation:
The coefficient of variation is given by:
[tex]\[
CV = \left(\frac{\sigma}{\mu}\right) \times 100
\][/tex]
where [tex]\( \sigma \)[/tex] is the standard deviation and [tex]\( \mu \)[/tex] is the mean.
3. Rearrange the Formula to Solve for Standard Deviation:
To find the standard deviation, we rearrange the formula:
[tex]\[
\sigma = CV \times \mu
\][/tex]
Replace [tex]\( CV \)[/tex] with [tex]\( 0.09 \)[/tex] and [tex]\( \mu \)[/tex] with 40:
[tex]\[
\sigma = 0.09 \times 40
\][/tex]
[tex]\[
\sigma = 3.6
\][/tex]
4. Calculate Variance:
Variance ([tex]\( \sigma^2 \)[/tex]) is the square of the standard deviation:
[tex]\[
\text{Variance} = \sigma^2 = (3.6)^2
\][/tex]
[tex]\[
\text{Variance} = 12.96
\][/tex]
So, the variance of the distribution is [tex]\( 12.96 \)[/tex].
