3 years ago
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 tion earn (a) below 250,000; (b) between 260,00 and 300,000?
1 Answers
Best Answer
cleanbanditticketsdonedeal Staff answered 3 years ago
cleanbanditticketsdonedeal Staff answered 3 years ago
Let X be random variable that represents salrie of some pediatric physician, X ~ N (mu, sigma^2), and with Z=(X- mu)/ sigma standard normal we have:
0.25=P(X<180)=P(Z< (180- mu)/ (sigma)) -> P(Z<-0.675)=0.25
0.25=P(X>320)=P(Z< (320- mu)/ (sigma)) -> P(Z>0.675)=0.25
And from that:
(180- mu)/ (sigma)=-0.675 -> 180- mu=-0.675 sigma
(320- mu)/ (sigma)=0.675 -> 320- mu=0.675 sigma
If we sum both expression we get 180- mu+320- mu=0 -> 500=2 my -> mu=250 and given that
sigma= (70)/(0.675)=103.7
a) P(X<250)=P(Z< (250-250)/(103.7))=P(Z<0)= Phi(0)=0.5
b) P(260<X<300)=P ((260-250)/(103.7)<Z< (300-250)/(103.7))
=P ((10)/(103.7)<Z< (50)/(103.7))
=P(0.096<Z<0.482)=P(Z<0.482)-P(Z<0.096)
= Phi(0.482)- Phi(0.096)=0.15