Question

Suppose a random variable is normally distributed. The probabilities for 85 and 142 are 10% and 65%, respectively. Find the mean and standard deviation tp the nearest whole number.

Answer 1

Solution:

Let the mean and standard deviation be $\,u, \sigma$ respectively.

$X\sim N(\mu,\sigma)$

$$P(X<85)=P(Z<\dfrac{85-\mu}{\sigma})=0.1$$ $$\dfrac{85-\mu}{\sigma}\approx-1.2815516$$ $$P(X<142)=P(Z<\dfrac{142-\mu}{\sigma})=0.65$$ $$\dfrac{142-\mu}{\sigma}\approx0.385320$$ $$\dfrac{85-\mu}{-1.2815516}=\dfrac{142-\mu}{0.385320}$$ $$\mu=\dfrac{85(0.385320)+142(1.2815516)}{0.385320+1.2815516}$$ $$\mu\approx128.8237$$ $$\sigma=\dfrac{142-128.8237}{0.385320}$$ $$\sigma=34.1957$$

Answer:

$\mu=129, \sigma=34$