Respuesta :
Answer:
We can use the z score formula given by:
[tex] z= \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
If we use this formula we got:
[tex]z=\frac{21-20}{\frac{9.31}{\sqrt{18}}}= 0.456[/tex]
[tex]z=\frac{25-20}{\frac{9.31}{\sqrt{18}}}= 2.279[/tex]
And using a calculator, excel or the normal standard table and we have that:
[tex]P(0.456<Z<2.279) = P(Z<2.279)-P(Z<0.476)= 0.989-0.676=0.313[/tex]
Step-by-step explanation:
We assume this previous info: It is known that the amounts of time required for room-service delivery at a certain Marriott Hotel are Normally distributed with the average delivery time of 20 minutes.
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 delivery times of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(20,9.31)[/tex]
Where [tex]\mu=20[/tex] and [tex]\sigma=9.31[/tex]
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 can use the z score formula given by:
[tex] z= \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
If we use this formula we got:
[tex]z=\frac{21-20}{\frac{9.31}{\sqrt{18}}}= 0.456[/tex]
[tex]z=\frac{25-20}{\frac{9.31}{\sqrt{18}}}= 2.279[/tex]
And using a calculator, excel or the normal standard table and we have that:
[tex]P(0.456<Z<2.279) = P(Z<2.279)-P(Z<0.476)= 0.989-0.676=0.313[/tex]
