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SAMPLING DISTRIBUTION OF THE MEAN
Recall,
What Do We Expect of Sample Means?
 The values of the sample mean x vary from random
sample to random sample in a predictable way.
 The center of the distribution of the x values is at the
true mean  (for any sample size n).
 With a larger sample size n, the x values tend to be
closer to the true population mean  . That is, the x
values vary less around the true mean  . The
variation will also depend on how much variation there
 When the sample size n is large, the sample mean
can take on many possible values, so the random
variable x can be viewed as a continuous random
variable with a density curve as its model.
 When the sample size n is large, the distribution of
the sample mean can be modeled approximately with
a normal distribution.
 For any sample size (even small), if the distribution of
the original population is normal, the distribution of
the sample mean can be modeled with a normal
distribution.
Sampling Distribution of X the Sample Mean
If simple random samples of size n are taken from a population
having population mean  and population standard deviation ,
then the sampling distribution of the sample mean has the following
properties:
1.
X  
2.  X 

n
The average of all the possible sample mean
values is equal to the parameter  . In other
words, the sample mean X is an unbiased
estimator of  .
The standard deviation of all of the possible
sample mean values decreases as the sample
size n increases.
Note: Keep in mind the difference between

(the standard
deviation for the original population values) and
standard deviation of the sample means).

n
(the
3. If the original population is normally distributed, then for
any sample size n the distribution of the sample mean is also
normal with mean and standard deviation as given.
 

X is N   ,

n

4. If the original population is not necessarily normally
distributed, but n is “sufficiently” large, the distribution of
the sample mean is approximately normal with mean and
standard deviation, as given.
  
X is approx. N   ,

n

Sufficiently large means that the normality does not hold exactly for
any one sample size n, but as n increases, the distribution
eventually starts to look like a normal distribution. A sample size of
30 or more is often considered large enough. If there are extreme
outliers in the observed data, it is better to have a larger sample.
This last result is known as the central limit theorem, abbreviated
CLT.
Example Average Expenditure
Let X be the random variable representing the amount spent by
patients at a drug store. The distribution of the X variable is
skewed to the right with a mean of   $60 and a standard
deviation $35. We plan to take a simple random sample of size
n=100 expenditures and will consider the sample mean
expenditure X .
(a) What is the value of the mean, the standard deviation, and
the (approximate) shape of the distribution of X , the sample
mean expenditure?
Mean of the distribution of X = $60
Standard deviation of the distribution of X =
$35
100
=$3.5
Shape of the distribution of X = N($60, $3.5) because the
sample is large ( n  30),
the CLT applies.
(b) Consider the following probability statement:
35
35
P( 60 
 X  60 
)  0.68
100
100
This statement is (circle one) .Explain.
True.
False
Can’t tell.
Let’s Do It! 3
The length of human pregnancy is approximately normally
distributed with mean  =266 days and a standard deviation
 =16.
a. What is the probability that a random sample of 20
pregnancies has a mean gestational period of 260 days or
less?
b. What is the probability that a random sample of 50
pregnancies has a mean gestational period within 10 days of
the mean?
Let’s Do It! 4
The mean weight of 15-year-old males is 142 pounds, and the
standard deviation is 12.3 pounds. If a sample of 36 15-year-old
males is selected, find the probability that the mean of the
sample will be greater than 144.5 pounds. Based on you answer,
would you consider the group overweight?
The Usefulness of Sampling Distributions
Did you discover the usefulness of sampling distributions?
As you see, in real life, you will NOT KNOW the true
population parameter, because if you did know it, you would
not need to take a sample. You take just ONE SAMPLE, not
many samples as in constructing the empirical sampling
distribution of its estimator. One sample may cost thousands
of dollars or take a lot of time to select.
So, what are we doing in this section?
We were discovering the behavior of a sampling distribution of
an estimator in the case of simple random samples. We
have observed that the larger the sample size taken, the more
the distribution looks “normal” and also the more concentrated
about the “true parameter value.” This guarantees that the
larger the sample size that we take, the greater the
chance that the estimator will be close to the true
parameter value.
In general, we take a sample from a population because we do not
know the true parameter value. If we have some theory regarding the
value of the parameter, we can use the expected sampling distribution
and the observed data to assess if the theory is supported or not.
Homework
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all, 63,any
64, one
76, 116
We cannot
say217:
how
particular estimate is to the true
parameter value. However, we can calculate the probability that the
true parameter value will be contained in the range:
This “something” is called the margin of error, and
we will learn more about it in the next chapter.