### Math 225 Course Notes

### Section 6.2:
Confidence Interval for a Population Mean

The sampling distribution of
is approximately normal with mean
and standard deviation
by
the central limit theorem
for sufficiently large samples.
The logic
of constructing confidence intervals for
from sample data depends on understanding
the sampling distribution
of
.
An example at the end of the section
demonstrates the type of exercise you should be able to solve.

Using the general formula
(estimate) +/- (reliability coefficient)(standard error)

constructing a confidence interval is simply plugging into the formula

To construct a C% confidence interval,
the multiplier z* is chosen from the standard
normal distribution so that the area between
-z* and z* is C%.
Common choices are

Confidence Level | z*
------------------------------------
90% | 1.645
95% | 1.960
99% | 2.576

A 95% confidence interval expressed as 4.5 +/- 2.3
can be interpreted as
"I am 95% confident that the population mean is within 2.3
of the sample mean 4.5" or
"I am 95% confident that the population mean is between 2.2 and 6.8".

Even if the population is not normal,
the central limit theorem says that the shape
of the sampling distribution will be approximately normal
for sufficiently large samples.
For most practical situations in biology,
samples of size 25 or 30 are sufficiently large
for confidence intervals based on the normal distribution
to be valid.
If your sample size is small and noticeably nonnormal,
with extreme outliers, or strong skewness apparent in histograms,
*you should not use the formula in this section for constructing
confidence intervals*.

In studies from recent years,
the birthweights of infants born in Boston
had population standard deviations of 20.6 ounces.
40 infants were randomly sampled from recent births in this population,
and the mean of their birth weights was 114.0 ounces.
Give a 95% confidence interval for the population mean.
The sample size is large enough that it is reasonable to conclude
that the shape of the sampling distribution of
is approximately normal.
From previous studies,
it is reasonable to conclude that the standard error of this distribution
is
= 3.26.

Plugging into the formula
gives an answer 114.0 +/- 6.4,
where we used 1.96 for z.

This can interpreted as
"we are 95% confident that the unknown mean weight of all infants
in Boston is between 107.6 and 120.4 ounces".

Last modified: Feb 26, 1996

Bret Larget,
larget@mathcs.duq.edu