Question

According to the National Association of Colleges and Employers, the average starting salary for new college graduates in health sciences is $51,541. The average starting salary for new college graduates in business is $53,901 (National Association of Colleges and Employers website, January 5, 2015).

Assume that starting salaries are normally distributed and that the standard deviation for starting salaries for new college graduates in health sciences is $11,000. Assume that the standard deviation for starting salaries for new college graduates in business $15,000.

a. What is the probability that a new college graduate in business will earn a starting salary of at least $65,000?

b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least $65,000?

c. What is the probability that a new college graduate in health sciences will earn a starting salary of less than $40,000?

d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences?

Answer 1

a) 22.96% probability that a new college graduate in business will earn a starting salary of at least $65,000

b) 11.12% probability that a new college graduate in health sciences will earn a starting salary of at least $65,000

c) 14.69% probability that a new college graduate in health sciences will earn a starting salary of less than $40,000

d) He would have to earn $88,776.

Explanation:

Problems of normally distributed samples can be 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:

Health sciences:

`\mu = 51541, \sigma = 11000`

Business:

`\mu = 53901, \sigma = 15000`

a. What is the probability that a new college graduate in business will earn a starting salary of at least $65,000?

This is 1 subtracted by the pvalue of Z when X = 65000. So

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

`Z = (65000 - 53901)/(15000)`

`Z = 0.74`

`Z = 0.74` has a pvalue of 0.7704

1 - 0.7704 = 0.2296

22.96% probability that a new college graduate in business will earn a starting salary of at least $65,000

b. What is the probability that a new college graduate in health sciences will earn a starting salary of at least $65,000?

This is 1 subtracted by the pvalue of Z when X = 65000. So

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

`Z = (65000 - 51541)/(11000)`

`Z = 1.22`

`Z = 1.22` has a pvalue of 0.8888

1 - 0.8888 = 0.1112

11.12% probability that a new college graduate in health sciences will earn a starting salary of at least $65,000

c. What is the probability that a new college graduate in health sciences will earn a starting salary of less than $40,000?

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

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

`Z = (40000 - 51541)/(11000)`

`Z = -1.05`

`Z = -1.05` has a pvalue of 0.1469

14.69% probability that a new college graduate in health sciences will earn a starting salary of less than $40,000

d. How much would a new college graduate in business have to earn in order to have a starting salary higher than 99% of all starting salaries of new college graduates in the health sciences?

This is X when Z has a pvalue of 0.99. So it is X when Z = 2.325.

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

`2.325 = (X - 53901)/(15000)`

`X - 53901 = 2.325*15000`

`X = 88776`

He would have to earn $88,776.