Question

15 percent of the workers in grade A,30 percent workers in grade B and 55 percent of the workers in grade C, if you receive amount less than 25000 you are in grade C,if you receive more than 56000 you are in grade A,find the mean and standard deviation of the income distribution assuming it is normally distributed

Answer 1

We have:

$P(x < 25000) = 0.55$

$P(x > 56000) = 0.15$

$P(25000 \le x \le 56000) = 0.3$

Let $a$ is mean, $\sigma$ is standard deviation. Then

$\Phi \left( {\frac{{25000 - a}}{\sigma }} \right) - \Phi \left( { - \infty } \right) = 0.55 \Rightarrow \Phi \left( {\frac{{25000 - a}}{\sigma }} \right) + 0.5 = 0.55 \Rightarrow \Phi \left( {\frac{{25000 - a}}{\sigma }} \right) = 0.05$

$\Phi \left( \infty \right) - \Phi \left( {\frac{{56000 - a}}{\sigma }} \right) = 0.15 \Rightarrow 0.5 - \Phi \left( {\frac{{56000 - a}}{\sigma }} \right) = 0.15 \Rightarrow \Phi \left( {\frac{{56000 - a}}{\sigma }} \right) = 0.35$

$\Phi \left( {\frac{{56000 - a}}{\sigma }} \right) - \Phi \left( {\frac{{25000 - a}}{\sigma }} \right) = 0.3$

If $\Phi \left( {\frac{{25000 - a}}{\sigma }} \right) = 0.05$ then $\frac{{25000 - a}}{\sigma } = 0.13$.

If $\Phi \left( {\frac{{56000 - a}}{\sigma }} \right) = 0.35$ then $\frac{{56000 - a}}{\sigma } = 1.04$ .

We have the system:

$\left\{ \begin{matrix} \frac{{25000 - a}}{\sigma } = 0.13\\ \frac{{56000 - a}}{\sigma } = 1.04 \end{matrix} \right. \Rightarrow \left\{ {\begin{matrix} {\sigma = \frac{{25000 - a}}{{0.13}}}\\ {\frac{{56000 - a}}{{\frac{{25000 - a}}{{0.13}}}} = 1.04} \end{matrix}} \right. \Rightarrow \left\{ {\begin{matrix} {\sigma = \frac{{25000 - a}}{{0.13}}}\\ {7280 - 0.13a = 26000 - 1.04a} \end{matrix}} \right. \Rightarrow \left\{ {\begin{matrix} {0.91a = 18720}\\ {\sigma = \frac{{25000 - a}}{{0.13}}} \end{matrix} \Rightarrow \left\{ {\begin{matrix} {a = \frac{{144000}}{7}}\\ {\sigma = \frac{{31000}}{{91}}} \end{matrix}} \right.} \right.$

Answer: ${a = \frac{{144000}}{7}}, {\sigma = \frac{{31000}}{{91}}}$