Respuesta :
Answer:
[tex]\mu=90[/tex] and [tex]\sigma=25[/tex]
From the central limit theorem (n>30) we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
With:
[tex]\mu_{\bar x}= 90[/tex]
[tex]\sigma_{\bar X}= \frac{25}{\sqrt{75}}= 2.887[/tex]
Step-by-step 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".
Solution to the problem
Let X the random variable that represent the heights of a population, and for this case we know the following properties for X
Where [tex]\mu=90[/tex] and [tex]\sigma=25[/tex]
From the central limit theorem (n>30) we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
With:
[tex]\mu_{\bar x}= 90[/tex]
[tex]\sigma_{\bar X}= \frac{25}{\sqrt{75}}= 2.887[/tex]
