Question

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?

Answer 1

a)
`z = (40000- 47173)/(6364) = -1.127`

b)
`z = (50000- 47173)/(6364) = 0.444`

c)
`P(X>40000)=P((X-\mu)/(\sigma)>(40000-\mu)/(\sigma))=P(Z>(40000-47173)/(6364))=P(z>-1.127)`

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

`P(z>-1.127)=1-P(z<-1.127)=1- 0.130 = 0.870`

Explanation:

Previous concepts

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
`\bar X=47173` and
`s=6364`

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

The z score for 40000 would be:

`z = (40000- 47173)/(6364) = -1.127`

Part b

The z score for 50000 would be:

`z = (50000- 47173)/(6364) = 0.444`

Part c

We are interested on this probability

`P(X>40000)`

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

`z=(x-\mu)/(\sigma)`

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

`P(X>40000)=P((X-\mu)/(\sigma)>(40000-\mu)/(\sigma))=P(Z>(40000-47173)/(6364))=P(z>-1.127)`

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

`P(z>-1.127)=1-P(z<-1.127)=1- 0.130 = 0.870`