Answer :
Answer:
[tex]P(\bar X >4.75)=P(Z>\frac{4.75-4.25}{\frac{1.6}{\sqrt{64}}}=2.5)[/tex]
Using the complement rule and the normal standard table or excel we have this:
[tex] P(Z>2.5)=1-P(Z<2.5) = 1-0.994=0.006[/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 assume the distribution for X is given by:
[tex]X \sim N(4.25,1.6)[/tex]
Where [tex]\mu=4.25[/tex] and [tex]\sigma=1.6[/tex]
And we select a sample of n =64. Since the distribution for X is normal then the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
And we want to find the following probability:
[tex] P(\bar X >4.75)[/tex]
And we can use the z score given by this formula:
[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
[tex]P(\bar X >4.75)=P(Z>\frac{4.75-4.25}{\frac{1.6}{\sqrt{64}}}=2.5)[/tex]
Using the complement rule and the normal standard table or excel we have this:
[tex] P(Z>2.5)=1-P(Z<2.5) = 1-0.994=0.006[/tex]