Respuesta :
Answer:
[tex] P(\bar X >70)= P(Z>\frac{70-69}{\frac{4}{\sqrt{36}}})= P(Z>1.5)[/tex]
And for this case we can use the complement rule and the normal standard distribution of excel and we got:
[tex] P(Z>1.5)=1-P(Z,1.5) = 1-0.933=0.0668[/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 distribution for X is given by:
[tex]X \sim N(69,4)[/tex]
Where [tex]\mu=69[/tex] and [tex]\sigma=4[/tex]
And we select a sample size of n =70
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]
And we want to find this probability:
[tex] P(\bar X >70)[/tex]
And we can use the z score formula given by:
[tex] z= \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
And using this formula we got:
[tex] P(\bar X >70)= P(Z>\frac{70-69}{\frac{4}{\sqrt{36}}})= P(Z>1.5)[/tex]
And for this case we can use the complement rule and the normal standard distribution of excel and we got:
[tex] P(Z>1.5)=1-P(Z,1.5) = 1-0.933=0.0668[/tex]
