# The salaries of pediatric physicians are approximately normally distributed. If

The salaries of pediatric physicians are approximately normally distributed. If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320,000, what fraction earn (a) below 250,000; (b) between 260,00 and 300,000?

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Daniel Williams

Let X be random variable that represents salrie of some pediatric physician, $$\displaystyle{X}\sim{N}{\left(\mu,\sigma^{{2}}\right)}$$, and with $$\displaystyle{Z}=\frac{{{X}-\mu}}{\sigma}$$ standard normal we have:
$$\displaystyle{0.25}={P}{\left({X}{<}{180}\right)}={P}{\left({Z}{<}{\frac{{{180}-\mu}}{{\sigma}}}\right)}\Rightarrow{P}{\left({Z}{<}-{0.675}\right)}={0.25}$$
$$\displaystyle{0.25}={P}{\left({X}{>}{320}\right)}={P}{\left({Z}{<}{\frac{{{320}-\mu}}{{\sigma}}}\right)}\Rightarrow{P}{\left({Z}{>}{0.675}\right)}={0.25}$$
And from that:
$$\displaystyle{\frac{{{180}-\mu}}{{\sigma}}}=-{0.675}\Rightarrow{180}-\mu=-{0.675}\sigma$$
$$\displaystyle{\frac{{{320}-\mu}}{{\sigma}}}={0.675}\Rightarrow{320}-\mu={0.675}\sigma$$
If we sum both expression we get $$\displaystyle{180}-\mu+{320}-\mu={0}\Rightarrow{500}={2}{m}{y}\Rightarrow\mu={250}$$ and given that
$$\displaystyle\sigma={\frac{{{70}}}{{{0.675}}}}={103.7}$$
a) $$\displaystyle{P}{\left({X}{<}{250}\right)}={P}{\left({Z}{<}{\frac{{{250}-{250}}}{{{103.7}}}}\right)}={P}{\left({Z}{<}{0}\right)}=\Phi{\left({0}\right)}={0.5}$$
b) $$\displaystyle{P}{\left({260}{<}{X}{<}{300}\right)}={P}{\left({\frac{{{260}-{250}}}{{{103.7}}}}{<}{Z}{<}{\frac{{{300}-{250}}}{{{103.7}}}}\right)}$$
$$\displaystyle={P}{\left({\frac{{{10}}}{{{103.7}}}}{<}{Z}{<}{\frac{{{50}}}{{{103.7}}}}\right)}$$
$$\displaystyle={P}{\left({0.096}{<}{Z}{<}{0.482}\right)}={P}{\left({Z}{<}{0.482}\right)}-{P}{\left({Z}{<}{0.096}\right)}$$
$$\displaystyle=\Phi{\left({0.482}\right)}-\Phi{\left({0.096}\right)}={0.15}$$