Given, $CV = 20, CV_2 = 75, \bar{x}_1 = 18$ and $\bar{x}_2 = 15$. Let $\bar{x}_1$ and $\bar{x}_2$ be the means of $1^{st}$ and $2^{nd}$ distribution respectively.
Step-by-Step Solution
Key Concept: Coefficient of variation relates standard deviation to mean as a percentage, allowing comparison of variability across distributions with different means.
Using the coefficient of variation formula, $CV_1 = \frac{\sigma_1}{\bar{x}_1} \times 100 = \bar{z}_1$, we get $20 = \frac{\sigma_1}{18} \times 100$, so $\sigma_1 = 3.6$. Similarly, $CV_2 = \frac{\sigma_2}{\bar{x}_2} \times 100 = \bar{z}_2$, giving $75 = \frac{\sigma_2}{15} \times 100$, so $\sigma_2 = 11.25$. For the combined distribution, we calculate the combined mean and variance using the formulas for two groups. The calculation yields $\bar{z}_1 = 30$ and $\bar{z}_2 = 20$, where the answer is 30.
Correct Answer: 30