Answer to Question #145133 in Marketing for Liezel

Question

The annual salaries of employees in a large company are approximately normally distributed with a mean of $50,000 and a standard deviation of $20,000.
a) What percent of people earn less than $40,000?
b) What percent of people earn between $45,000 and $65,000?
c) What percent of people earn more than $70,000?

Answer 1

A. Let S be the random variable of a salary of employee (in $), S ~ N(50000,20000). Then the random 

variable X =𝑆−50000

20000

~N(0,1).

𝑃(𝑆 < 40000) = 𝑃 (𝑋 <

40000 − 50000

20000 ) = 𝑃(𝑋 < −0.5) = 𝛷(−0.5) = 0.3085375.

Here Φ(x) denotes the cumulative distribution function of a standard normal distribution.

Answer: 31%.

b. What percent of people earn between $45000 and $65000?

Solution: 

𝑃(45000 < 𝑆 < 65000) = 𝑃 (

45000 − 50000

20000 < 𝑋 <

65000 − 50000

20000 ) = 𝑃(−0.25 < 𝑋 < 0.75)

= 𝛷(0.75) − 𝛷(−0.25) = 0.7733726 − 0.4012937 = 0.3720789.

Answer: 37%.

c. What percent of people earn more than $70000?

Solution: 

𝑃(𝑆 > 70000) = 𝑃 (𝑋 >

70000 − 50000

20000 ) = 𝑃(𝑋 > 1) = 0.8413447.

Answer: 84%.