Asimple random sample of size nequals=8181 is obtained from a population with mu equals 77μ=77 and sigma equals 27σ=27. (a) describe the sampling distribution of x overbarx. (b) what is upper p (x overbar greater than 81.5 )p x> 81.5? (c) what is upper p (x overbar less than or equals 69.5 )p x≤69.5? (d) what is upper p (73.4 less than x overbar less than 84.05 )p 73.4

a)

b)

c)  

Step-by-step explanation:

1) 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 interest on this case, and for this case we know the distribution for X is given by:

And let represent the sample mean, the distribution for the sample mean is given by:

On this case  

Part a

We want this probability:

The best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

Part b

We want this probability:

If we apply the formula for the z score to our probability we got this:

Part c

We are interested on this probability

If we apply the Z score formula to our probability we got this:

And we can find this probability on this way:

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

unless if u could do my questions, if not i wouldn't want to