Download Section 7.3

Section 7.3 The Sampling Distribution of the Sample Mean X
This section focuses on when a variable under consideration is
normally distributed or is not normally distributed and how that affects the
sampling distribution.
Key Fact 7.2
Sampling Distribution of the Sample Mean for a Normally Distributed
Variable
Suppose that a variable X of a population is normally distributed with mean

 . Then for samples of size n the variable X
is also normally distributed and has mean  and standard deviation

and standard deviation
n
Example
The distribution for a 10 kilometer run in New York City is normally
distributed. The mean time to finish the race is
standard deviation is
  61 minutes and the
  9 minutes .
Find the following:
a. The population distribution
b. The sampling distribution of the sample mean for samples of size 4
c. The sampling distribution of the sample mean for samples of size 9
d. What do you notice about the population distribution and the two
sampling distributions?
We have been focusing on normal distributions and how changing the
sample size for a sampling distribution can affect the shape.
The next question to ask is how a non-normal distribution is affected by
sampling distributions of various sizes
Why don’t we look at sampling distributions of size 10, 20 and 30 if the
variable under consideration’s shape is uniform or is j-shaped.
These results bring us to the Center Limit Theorem
Key Fact 7.3
Central Limit Theorem
For a relatively large sample size the variable
X is approximately
normally distributed, regardless of the distribution of the variable
under consideration. This approximation becomes better with
increasing sample size.
Example 1
a. For 10 kilometer New York City race, what is the probability that the
sampling error made in estimation the population mean finishing time
by that of a random sample of four runners will be no more than five
minutes off from the true population mean time. The mean time to
finish the race is
  61 minutes and the standard deviation is
  9 minutes .
b. If the random sample is changed from four runners to nine runners,
what is the probability that the sampling error made in estimation the
population mean finishing time by that of a random sample of nine
runners will be no more than five minutes off from the true population
mean time.
Example 2
Key Fact 7.4
Sampling Distribution of the Sample Mean
Suppose that a variable X of a population has mean
deviation  . Then, for samples of size n
 The mean of
 and standard
X equals the population mean, or  X  
 The standard deviation of
X equals the population standard
deviation divided by the square root of the sample size, or
X 
 If

n
X is normally distributed, so is X , regardless of sample
size; and
 If the sample size is large
X is approximately normally
distributed, regardless of the distribution of X