A researcher collected data from a sample of correctional officers and found that on average they made 47,173 dollars a year with S=6,364. (10 points) a. Calculate the z-score for a correctional officer who makes 40,000 dollars a year b. Calculate the z-score for a correctional officer who makes 50,000 dollars a year? c. What proportion of correctional officers make more than 40,000 dollars a years?

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Part a

Let X the random variable that represent the amount of money of a population of interet, and we know the following info:

Where [tex]\bar X=47173[/tex] and [tex]s=6364[/tex]

We can assume that the sample size is large and the estimators can be used as a good description for the parameters [tex]\mu \sigma[/tex]

The z score for 40000 would be:

[tex]z = \frac{40000- 47173}{6364} = -1.127[/tex]

Part b

The z score for 50000 would be:

[tex]z = \frac{50000- 47173}{6364} = 0.444[/tex]

Part c

We are interested on this probability

[tex]P(X>40000)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X>40000)=P(\frac{X-\mu}{\sigma}>\frac{40000-\mu}{\sigma})=P(Z>\frac{40000-47173}{6364})=P(z>-1.127)[/tex]

And we can find this probability using the complement rule and the normal standard distribution or excel and we got:

[tex]P(z>-1.127)=1-P(z<-1.127)=1- 0.130 = 0.870[/tex]