31 The salaries of pediatric physicians are approximately no

31. 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?

Solution

Mean of normal distribution = (180000+320000)/2 = 250,000

a)

For a normal distribution half of them will be below mean,

So, fraction with salaries below 250000 = 1/2 or 50%

b)

Let variance be sigma^2

P( salary < 180000) = 0.25

z score for this salary = -0.674

-0.674 = (180,000-250,000)/sigma

sigma = (180,000-250,000)/( -0.674) = 103857.6

P( 260000 < salary < 300000) = P( (260,000-250,000)/103857.6 < z < (300,000-250,000)/103857.6 )

= P( 0.096286 < z< 0.481428 ) = 0.68489-0.53835 =0.14654 = 14.654 %