a) Find the probability an individual large-cap domestic stock fund had a three-year return of 10% or less.
b) How big does the return have to be to put a domestic stock fund in the top 25% for the three-year period?
Answer :
Given :
The population mean $(μ) = 14.4$
The population standard deviation $(σ) = 4.4$
Solution :
a) The probability an individual large-cap domestic stock fund had a three-year return of 10% or less :
$$\text{P}(x \leq 10) = \text{P}\left(\frac{x – \mu}{\sigma} \leq \frac{10 – 14.4}{4.4}\right)$$ $$= \text{P}(z \leq -1)$$ $$= 0.1587$$
b) Let us assume that the return have to be $x_0$ % to put a domestic stock fund in the top 25% for the three-year period.
$$\therefore \text{P}(x > x_0) = 0.25$$ $$\therefore 1 – \text{P}(x > x_0) = 1 – 0.25$$ $$\therefore \text{P}(x < x_0) = 0.75$$ $$\therefore \text{P} \left( z < \frac{x_0 – \mu}{\sigma} \right) = 0.75$$ $$\therefore \frac{x_0 – \mu}{\sigma} = 0.674$$ $$\therefore \frac{x_0 – 14.4}{4.4} = 0.674$$ $$\therefore x_0 – 14.4 = 0.674 \times 4.4$$ $$\therefore x_0 \approx 17.37\%$$
