Answer:
There is a 34.15% probability the crew member earns between $20.50 and $24.00 per hour
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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
.
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 = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{24 - 20.50}{3.50}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B24%20-%2020.50%7D%7B3.50%7D)
![Z = 1](https://tex.z-dn.net/?f=Z%20%3D%201)
has a pvalue of 0.8413.
X = 20.50
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{20.50 - 20.50}{3.50}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B20.50%20-%2020.50%7D%7B3.50%7D)
![Z = 0](https://tex.z-dn.net/?f=Z%20%3D%200)
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