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# Answer to Question #145133 in Marketing for Liezel

Question #145133
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?
1
2020-11-18T06:00:52-0500

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.

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