### Math 225 Course Notes

### Section 5.3: Distribution of the Sample Mean

The sampling distribution of
can be described by its
center,
spread,
and shape.
For most problems,
the central limit theorem
allows us to conclude that the shape of the distribution
is approximately normal,
and we can use methods from chapter 4
to answer questions about the probability of the sample mean
falling in various intervals.
An example at the end of the section
demonstrates the type of exercise you should be able to solve.

The mean of the sampling distribution of
is the population mean
.
This is the balancing point of the sampling distribution.

The standard deviation of the sampling distribution of
,
also know as the *standard error* of
,
or SE(),
is smaller than the population standard deviation.
This reflects the fact that the mean from a sample
is more likely to be close
to the population mean
than a randomly chosen individual.
The formula for SE() is
.
The standard error may usually be thought of as a typical size
for the distance between
and
due to chance.

The shape of the sampling distribution of
will be approximately normal (bell-shaped) for sufficiently large samples.
For many sets of data arising in practice,
"sufficiently large" might mean 25 or 30.
Populations where "sufficiently large" needs to be substantially larger
than 25 or 30
will generally be strongly skewed.

The central limit theorem summarizes the information
about the center, spread, and shape presented above.

*For every population
with a population mean
and population standard deviation
,
even if the shape of the population is not normal,
the sampling distribution of
will be approximately normal
with a mean of
and a standard deviation of
for large samples.
*

In a recent year,
the birthweights of infants born in Boston
had a mean weight of 112.0 ounces
with a standard deviation of 20.6 ounces.
If 40 infants were randomly sampled,
what is the probability that the mean of their weights would be
between 105 and 115 ounces?
Draw a sketch!

By the central limit theorem,
the sampling distribution of
should be approximately normally distributed with a mean of
= 112.0
and an SE of
= 3.26.
The two border points have z-scores of

z = (105 - 112)/3.26 = -2.15 and z = (115 - 112)/3.26 = 0.92.

The area between these points on the standard normal curve is

.8212 - .0158 = .8054, so there is about an 80% chance that the sample
mean would be between 105 and 115.

Last modified: Feb 19, 1996

Bret Larget,
larget@mathcs.duq.edu