Confidence Interval for the Mean
David M. Lane
Under Normal Distributions, Sampling
Distribution of the Mean, Introduction
to Estimation, Introduction
to Confidence Intervals
- Use the normal distribution calculator to find the value of z to use
for a confidence interval
- Compute a confidence interval on the mean when σ is
- Determine whether to use a t distribution or a normal distribution
- Compute a confidence interval on the mean when σ is estimated
When you compute a confidence interval, you compute
the mean of a sample in order to estimate the mean of the population.
Clearly, if you already knew the population mean, there would
be no need for a confidence interval.
Recall from the section on the sampling distribution
of the mean that the mean of the sampling
distribution is μ and the standard
error of the mean is
In general, you compute the 95% confidence interval
for the mean with the following formula:
Lower limit = M - Z.95σm
Upper limit = M + Z.95σm
where Z.95 is the number of standard
deviations extending from the mean of a normal distribution required
to contain 0.95 of the area and σm
is the standard error of the mean.
If you look closely at this formula for a confidence
interval, you will notice that you need to know the standard deviation (σ)
in order to estimate the mean. This may sound unrealistic, and
it is. However, computing a confidence interval when σ is
known is easier than when σ has to be estimated, and serves
a pedagogical purpose. Later in this section we will show how
to compute a confidence interval for the mean when σ has
to be estimated.
When the variance is not known but has to be estimated
from sample data you should use the t distribution rather than
the normal distribution. When the sample size is large, say 100
or above, the t distribution is very similar to the standard normal
distribution. However, with smaller sample sizes, the t distribution
is leptokurtic, which means it has relatively more scores in its
tails than does the normal distribution. As a result, you have
to extend farther from the mean to contain a given proportion
of the area. Recall that with a normal distribution, 95% of the
distribution is within 1.96 standard deviations of the mean. If
you have a sample size of only 5, 95% of the area is within 2.78
standard deviations of the mean. Therefore, the standard error
of the mean would be multiplied by 2.78 rather than 1.96.
The values of t to be used in a confidence interval
can be looked up in a table of the t distribution.
You can also use the "inverse
t distribution" calculator to find the t values to use
in confidence intervals. You will learn more about the t distribution
in the next section.
More generally, the formula for the 95% confidence
interval on the mean is:
Lower limit = M - (tCL)(sm)
Upper limit = M + (tCL)(sm)
where M is the sample mean, tCL
is the t for the confidence level desired (0.95 in the above example),
and sm is the estimated standard error
of the mean.
Please answer the questions: