Question

The salaries of physicians in a certain specialty are approximately normally distributed. If 25 percent of these physicians earn less than $180,000 and 25 percent earn more than $320,000, approximately what fraction earn

(a) less than $200,000?

(b) between $280,000 and $320,000?

Answer 1

Solution:-

a)

x = 180,000

p-value for the lowest 25% = 0.25

z-score for the p-value = - 0.675

By applying normal distribution:-

-0.675 = 180,000 - u

u = 180,000 + 0.675

x = 320,000

p-value for the top 25% = 0.75

z-score for the p-value = 0.675

By applying normal distribution:-

0.675 = 320,000 - u

u = 320,000 - 0.675

180,000 + 0.675 = 320,000 - 0.675

1.35 ​ = 140000

= 103703.7

u = 250,000

a) The fraction that earn less than $200,000 is 0.423.

x = 200,000

By applying normal distribution:-

z = 0.193

P(z > 0.193) = 0.423

b) The fraction that earn between $280,000 and $320,000 is 0.136.

x₁ = 280,000

x₂ = 320,000

By applying normal distribution:-

z₁ = 0.289

z₂ = 0.675

P( 0.289 < z < 0.675) = P(z > 0.289) - P(z > 0.675)  

P( 0.289 < z < 0.675) = 0.386 - 0.25

P( 0.289 < z < 0.675) = 0.136