Answer:
a)
![z = (40000- 47173)/(6364) = -1.127](https://img.qammunity.org/2021/formulas/mathematics/college/euszpuuvhpm5v8crwfwqjk8j2mhbb05b61.png)
b)
![z = (50000- 47173)/(6364) = 0.444](https://img.qammunity.org/2021/formulas/mathematics/college/aric9s3jcnreu56qufnvru69c0fmdebzuu.png)
c)
And we can find this probability using the complement rule and the normal standard distribution or excel and we got:
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
and
We can assume that the sample size is large and the estimators can be used as a good description for the parameters
![\mu \sigma](https://img.qammunity.org/2021/formulas/mathematics/college/ow9iyl6ztv0eduhlsvd53bmya6gyfprc9c.png)
The z score for 40000 would be:
![z = (40000- 47173)/(6364) = -1.127](https://img.qammunity.org/2021/formulas/mathematics/college/euszpuuvhpm5v8crwfwqjk8j2mhbb05b61.png)
Part b
The z score for 50000 would be:
![z = (50000- 47173)/(6364) = 0.444](https://img.qammunity.org/2021/formulas/mathematics/college/aric9s3jcnreu56qufnvru69c0fmdebzuu.png)
Part c
We are interested on this probability
And the best way to solve this problem is using the normal standard distribution and the z score given by:
If we apply this formula to our probability we got this:
And we can find this probability using the complement rule and the normal standard distribution or excel and we got: