Answer to Question #275527 in Statistics and Probability for Itachi

Question

Suppose a random variable is normally distributed. The probabilities for a85 and 142 are 10% and 65%, respectively. Find the mean and standard deviation to the nearest whole number.

Answer 1

$$P(X<85)=P(Z<\dfrac{85-\mu}{\sigma})=0.1$$

$$\dfrac{85-\mu}{\sigma}\approx-1.2815516$$

$$P(X<142)=P(Z<\dfrac{142-\mu}{\sigma})=0.65$$

$$\dfrac{142-\mu}{\sigma}\approx0.385320$$

$$\dfrac{85-\mu}{-1.2815516}=\dfrac{142-\mu}{0.385320}$$

$$\mu=\dfrac{85(0.385320)+142(1.2815516)}{0.385320+1.2815516}$$

$$\mu\approx128.8237$$

$$\sigma=\dfrac{142-128.8237}{0.385320}$$

$$\sigma=34.1957$$

Answer:

$\mu=129, \sigma=34$