Question

The annual wages X of the staffs of collage of natural sciences are normally

distributed with mean µ and standard deviation σ. Knowing that P(X > 42320) =

0.0853 and P(X < 41230) = 0.6103, find µ and σ.

Answer 1

$P(X>42320)=0.0853;\\ P(X<42320)=P(Z<z_1)=\\ =1-0.0853=0.9147;\\ P(X<41230)=0.6103;\\ z_1=\cfrac{42320-\mu}{\sigma};\\ z_2=\cfrac{41230-\mu}{\sigma}.$

From z-table we find $P(1.37)=0.9147,P(0.28)=0.6103.$

So,

$\begin{cases} \cfrac{42320-\mu}{\sigma}=1.37\\ \cfrac{41230-\mu}{\sigma}=0.28 \end{cases}$

$\begin{cases} 42320-\mu=1.37\sigma\\ 41230-\mu=0.28\sigma \end{cases}$

$\begin{cases} \mu=42320-1.37\sigma\\ \mu=41230-0.28\sigma \end{cases}$

$42320-1.37\sigma=41230-0.28\sigma\\ 1.09\sigma=1090\\ \sigma=1000\\ \mu=41230-0.28\cdot1000=40950.$