Answer:
a)
Because the sample mean is defined as:
So then the mean for the sample distibution is
b)
c) For this case since the distribution for the random variable is normal and for the sample mean is also normal so then we can conclude that the distribution for the sample mean is symmetrical and bell shaped with a deviation lower than the deviation for X
Explanation:
Assuming this complete problem:
The customer care manager at a cell phone company keeps track of how long each help-line caller spends on hold before speaking to a customer service representative. He finds that the distribution of wait times for all callers has a mean of 12 minutes with a standard deviation of 5 minutes. The distribution is moderately skewed to the right. Suppose the manager takes a random sample of 10 callers and calculates their mean wait time, .
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".
Let X the random variable that represent the wait times for all callers of a population, and for this case we know the distribution for X is given by:
Where and
(a) What is the mean of the sampling distribution of
?
Since the distribution for the random variable X is normal then the distribution for the random variable is also normal and given by:
Because the sample mean is defined as:
So then the mean for the sample distibution is
(b) Is it possible to calculate the standard deviation of ? If it is, do the calculation. If it isn’t, explain why.
For this case since we knwo that the distribution for the sampel mena is given by:
Then we know that the deviation for the sample mean is given by:
(c) Do you know the approximate shape of the sampling distribution of ? If so, describe the shape and justify your answer. If not, explain why not.
For this case since the distribution for the random variable is normal and for the sample mean is also normal so then we can conclude that the distribution for the sample mean is symmetrical and bell shaped with a deviation lower than the deviation for X