Download Section 7.2 Mean and Standard Deviation of X distributed and we`ll

Section 7.2 Mean and Standard Deviation of X
In general we do not know the exact sampling distribution of the
sample mean exactly.
If we have a population size of 18 and want to see all of the samples of
size 10, we would have 43,758 samples. This number of samples is way to
large to even consider.
We can often approximate a sampling distribution by a normal
distribution. Under certain conditions the variable X is normally
distributed and we’ll discuss what those “certain conditions” are.
One condition is when the variable X is normally distributed. If X is
normally distributed, then X is normally distributed.
Mean of the variable X
For samples of size n, the mean of the variable X equals the mean of
the variable under consideration.
X  
Example
If we go back and look at the billionaire data for the mean of the 15 samples
of size 2 or the mean the 6 samples of size 5, we will see that both have the
same mean as the population mean.
Standard deviation of the variable X
If the sampling is done with our replacement from a finite population and
n  0.05  N where n is the sample size and N is the population size then
the standard deviation for the variable X is
X 
N n 

N 1 n
But if n  0.05  N then the equation to use is
X 

n
This equation is the one that is most commonly used.
Example
The billionaire example falls under the category of n  0.05  N so we can
find the standard deviation of the population and the standard deviation of
the samples of size 2 or the samples of size 5 to illustrate this relationship.
Summary
 The larger that the sample size is the smaller that the standard
deviation of X will be.
 The smaller the standard deviation of X the more closely the
possible values of X (the possible sample means) will cluster
around the mean of X
 The mean of all of the X equals the population mean
X  