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Chapter 18: Sampling Distribution of Sample Means, Ç Known
Big Ideas: Averages are less variable than individual observations
Averages are more normally distributed than individual observations.
Suppose that x is the mean of an SRS of size n drawn from a large population with mean
standard deviation
deviation is

x



x
x
n
. Then the mean of the sampling distribution of x is
x 
x

x
and
and its standard
.
Central Limit Theorem
Start with a population with a given mean

x
, a standard deviation

x
, and any shape distribution. Pick
a sufficiently large sample size n (at least 30 is a good rule of thumb) and take all possible samples of
size n. Compute the mean of each of these samples.
1. The set of all sample means is approximately normally distributed.
2. The mean of the set of sample means equals the mean of the population (i.e.,
 x   x)
3. The standard deviation of the set of sample means is approximately equal to the standard deviation of
the population divided by the square root of the sample size (i.e.,

x


x
n
)
The population needs to be at least 10x the sample size for this formula to be valid.
Central Limit Theorem (condensed version): If one takes a simple random sample of size n from a
population with mean  x and standard deviation  x , then as n gets larger, the distribution of
x approaches a normal distribution with mean 
x
and standard deviation
x
n
.
If the original distribution is normal, then we can conclude that the set of sample means has a normal
distribution, regardless of the sample size.
No matter what the shape of the original distribution (normal or otherwise), if n is large enough, then the
set of sample means is approximately normally distributed.
The Central Limit Theorem focuses on the shape of the sampling distribution. The statements regarding
the mean and standard deviation of the sampling distribution are true regardless of the shape of the
sampling distribution.
The mean of the sampling distribution of x is an unbiased estimator of the population mean
.
Example 1: Suppose that tomatoes weigh an average of 10 ounces with a standard deviation of 3 ounces.
A store sells boxes containing 12 tomatoes each. If customers determine the average weight of one
tomato for each box they buy, what will be the mean and standard deviation of these average weights?
Example 2: Suppose that the distribution for total amounts spent by students vacationing for a week in
Florida is normally distributed with a mean of $650 and a standard deviation of $120.
a) What is the probability that a student will spend between $600 and $700 during the week?
b) What is the probability that a group of 10 students will spend an average of between $600 and $700
during the week?
Example 3: Suppose that the average outstanding credit card balance for young couples is $1250 with a
standard deviation of $420. If 100 couples are selected at random, what is the probability that the
mean outstanding credit card balance exceeds $1300?