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) 30.85% of people earn less than $40,000

b) 37.21% of people earn between $45,000 and $65,000.

c) 15.87% of people earn more than $70,000

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 50000, \sigma = 20000`

a.What percent of people earn less than $40,000?

This is the pvalue of Z when X = 40000. So

`Z = (X - \mu)/(\sigma)`

`Z = (40000 - 50000)/(20000)`

`Z = -0.5`

`Z = -0.5` has a pvalue of 0.3085.

30.85% of people earn less than $40,000

b.What percent of people earn between $45,000 and $65,000?

This is the pvalue of Z when X = 65000 subtracted by the pvalue of Z when X = 45000. So

X = 65000

`Z = (X - \mu)/(\sigma)`

`Z = (65000 - 50000)/(20000)`

`Z = 0.75`

`Z = 0.75` has a pvalue of 0.7734.

X = 45000

`Z = (X - \mu)/(\sigma)`

`Z = (45000 - 50000)/(20000)`

`Z = -0.25`

`Z = -0.25` has a pvalue of 0.4013.

0.7734 - 0.4013 = 0.3721

37.21% of people earn between $45,000 and $65,000.

c.What percent of people earn more than $70,000?

This is 1 subtracted by the pvalue of Z when X = 70000. So

`Z = (X - \mu)/(\sigma)`

`Z = (70000 - 50000)/(20000)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413.

1 - 0.8413 = 0.1587

15.87% of people earn more than $70,000