Question

A recent study of the hourly wages of maintenance crew members for major airlines showed that the mean hourly salary was $20.50, with a standard deviation of $3.50. Assume the distribution of hourly wages follows the normal probability distribution. We select a crew member at random.

What is the probability the crew member earns:


(a) Between $20.50 and $24.00 per hour (Round your answer to 4 decimal places.)


Probability

Answer 1

There is a 34.15% probability the crew member earns between $20.50 and $24.00 per hour

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:

A recent study of the hourly wages of maintenance crew members for major airlines showed that the mean hourly salary was $20.50, with a standard deviation of $3.50. This means that
`\mu = 20.50, \sigma = 3.50`.

a) What is the probability the crew member earns between $20.50 and $24.00 per hour?

This is the pvalue of the zscore of X = 24.00 subtracted by the pvalue of the zscore of X = 20.50.

X = 24

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

`Z = (24 - 20.50)/(3.50)`

`Z = 1`

`Z = 1` has a pvalue of 0.8413.

X = 20.50

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

`Z = (20.50 - 20.50)/(3.50)`

`Z = 0`

`Z = 0` has a pvalue of 0.50.

This means that there is a 0.8413 - 0.50 = 0.3413 = 34.15% probability the crew member earns between $20.50 and $24.00 per hour