Answer:
a)
b)
Using the complement rule and the normal standard table or excel we have:
c)
We can find this probability with the following operation:
![P(-1.549< Z< 1.549) = P(Z](https://tex.z-dn.net/?f=%20P%28-1.549%3C%20Z%3C%201.549%29%20%3D%20P%28Z%3C1.549%29-P%28Z%3C-1.549%29%20%3D%200.939-0.061%3D%200.879)
d)
So on this case the 95% confidence interval would be given by (1623.48;1876.517)
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".
Part a
Let X the random variable that represent the volume of overnight bags of a population, and for this case we know the distribution for X is given by:
Where
and
Since the distribution for X is normal the distribution for the sample mean
is given by:
Part b
For this case we want this probability:
![P(\bar X >1800)](https://tex.z-dn.net/?f=%20P%28%5Cbar%20X%20%3E1800%29)
We can use the z score given by:
![z= \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}](https://tex.z-dn.net/?f=%20z%3D%20%5Cfrac%7B%5Cbar%20X%20-%5Cmu%7D%7B%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D%7D)
Using this formula we got:
Using the complement rule and the normal standard table or excel we have:
Part c
For this case we want this probability:
![P(1650](https://tex.z-dn.net/?f=%20P%281650%3C%5Cbar%20X%20%3C1850%29)
Using the z score formula we got:
We can find this probability with the following operation:
![P(-1.549< Z< 1.549) = P(Z](https://tex.z-dn.net/?f=%20P%28-1.549%3C%20Z%3C%201.549%29%20%3D%20P%28Z%3C1.549%29-P%28Z%3C-1.549%29%20%3D%200.939-0.061%3D%200.879)
Part d
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
The confidence interval for the mean is given by the following formula:
(1)
Since the Confidence is 0.95 or 95%, the value of
and
, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.025,0,1)".And we see that
Now we have everything in order to replace into formula (1):
So on this case the 95% confidence interval would be given by (1623.48;1876.517)