Answer and explanation:
Given : Foot locker uses sales per square foot as a measure of store productivity. sales are currently running at an annual rate of $406 per square foot. you have been asked by management to conduct a study of a sample of 64 foot locker stores.Assume the standard deviate in annual sales per square foot for the population of all 3400 Foot locker stores is $80.
To find :
a) Show the sampling distribution of x bar, the sample mean annual sales per square foot for a sample of 64 Foot locker stores.
By the central limit theorem also specifies that
will have the same expected value as the population mean, that is ![E(\bar{x}) = 406](https://tex.z-dn.net/?f=E%28%5Cbar%7Bx%7D%29%20%3D%20406)
The standard deviation of
is given by,
![\sigma_{\bar{x}}=\frac{\sigma}{\sqrt n}](https://tex.z-dn.net/?f=%5Csigma_%7B%5Cbar%7Bx%7D%7D%3D%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%20n%7D)
![\sigma_{\bar{x}}=\frac{80}{\sqrt {64}}](https://tex.z-dn.net/?f=%5Csigma_%7B%5Cbar%7Bx%7D%7D%3D%5Cfrac%7B80%7D%7B%5Csqrt%20%7B64%7D%7D)
![\sigma_{\bar{x}}=\frac{80}{8}](https://tex.z-dn.net/?f=%5Csigma_%7B%5Cbar%7Bx%7D%7D%3D%5Cfrac%7B80%7D%7B8%7D)
![\sigma_{\bar{x}}=10](https://tex.z-dn.net/?f=%5Csigma_%7B%5Cbar%7Bx%7D%7D%3D10)
b) What is the probability that the sample mean will be within $15 of the population mean?
The probability is given by, ![P(-15](https://tex.z-dn.net/?f=P%28-15%20%3Cx-%5Cbar%7Bx%7D%20%3C15%29)
![P(\frac{-15}{\sigma_{\bar{x}}}](https://tex.z-dn.net/?f=P%28%5Cfrac%7B-15%7D%7B%5Csigma_%7B%5Cbar%7Bx%7D%7D%7D%20%3C%5Cfrac%7Bx-%5Cbar%7Bx%7D%7D%7B%5Csigma_%7B%5Cbar%7Bx%7D%7D%7D%20%3C%5Cfrac%7B15%7D%7B%5Csigma_%7B%5Cbar%7Bx%7D%7D%7D%29)
![P(\frac{-15}{10}](https://tex.z-dn.net/?f=P%28%5Cfrac%7B-15%7D%7B10%7D%20%3C%5Cfrac%7Bx-%5Cbar%7Bx%7D%7D%7B%5Csigma_%7B%5Cbar%7Bx%7D%7D%7D%20%3C%5Cfrac%7B15%7D%7B10%7D%29)
![P(-1.5](https://tex.z-dn.net/?f=P%28-1.5%3CZ%3C1.5%29)
From standard normal table,
![P(-1.5](https://tex.z-dn.net/?f=P%28-1.5%3CZ%3C1.5%29%3D0.8664)
c) Suppose you find a sample mean of $380. what is the probability of finding a sample mean of $380 or less? would you consider such a sample to be unusually low performing group of stores?
![P(\bar{x}](https://tex.z-dn.net/?f=P%28%5Cbar%7Bx%7D%3C380%29%3DP%28Z%3C%5Cfrac%7Bx-%5Cbar%7Bx%7D%7D%7B%5Csigma_%7B%5Cbar%7Bx%7D%7D%7D%29)
![P(\bar{x}](https://tex.z-dn.net/?f=P%28%5Cbar%7Bx%7D%3C380%29%3DP%28Z%3C%5Cfrac%7B380-406%7D%7B10%7D%29)
![P(\bar{x}](https://tex.z-dn.net/?f=P%28%5Cbar%7Bx%7D%3C380%29%3DP%28Z%3C%5Cfrac%7B-26%7D%7B10%7D%29)
![P(\bar{x}](https://tex.z-dn.net/?f=P%28%5Cbar%7Bx%7D%3C380%29%3DP%28Z%3C-2.6%29)
From standard normal table,
![P(\bar{x}](https://tex.z-dn.net/?f=P%28%5Cbar%7Bx%7D%3C380%29%3D0.0047)