State why a confidence interval is not the probability the interval
contains the parameter

Say you were interested in the mean weight of
10-year-old girls living in the United States. Since it would
have been impractical to weigh all the 10-year-old girls in the
United States, you took a sample of 16 and found that the mean
weight was 90 pounds. This sample
mean of 90 is a point
estimate of the population
mean. A point estimate by itself is of limited usefulness because
it does not reveal the uncertainty associated with the estimate;
you do not have a good sense of how far this sample mean may be
from the population mean. For example, can you be confident that
the population mean is within 5 pounds of 90? You simply do not
know.

Confidence intervals provide more information than
point estimates. Confidence intervals for means are intervals
constructed using a procedure (presented in the next
section)
that will contain the population mean a specified proportion of
the time, typically either 95% or 99% of the time. These intervals
are referred to as 95% and 99% confidence intervals respectively.
An example of a 95% confidence interval is shown below:

72.85 < μ < 107.15

There is good reason to believe that the population
mean lies between these two bounds of 72.85 and 107.15 since 95%
of the time confidence intervals contain the true mean.

If repeated samples were taken and the 95% confidence
interval computed for each sample, 95% of the intervals would
contain the population mean. Naturally, 5% of the intervals would
not contain the population mean.

It is natural to interpret a 95% confidence interval
as an interval with a 0.95 probability of containing the population
mean. However, the proper interpretation is not that simple. One
problem is that the computation of a confidence interval does
not take into account any other information you might have about
the value of the population mean. For example, if numerous prior
studies had all found sample means above 110, it would not make
sense to conclude that there is a 0.95 probability that the population
mean is between 72.85 and 107.15. What about situations in which
there is no prior information about the value of the population
mean? Even here the interpretation is complex. The problem is
that there can be more than one procedure that produces intervals
that contain the population parameter 95% of the time. Which procedure
produces the "true" 95% confidence interval? Although
the various methods are equal from a purely mathematical point
of view, the standard method of computing confidence intervals
has two desirable properties: each interval is symmetric about
the point estimate and each interval is contiguous. Recall from
the introductory
section in the chapter on probability that, for some purposes,
probability is best thought of as subjective. It is reasonable,
although not required by the laws of probability, that one adopt
a subjective probability of 0.95 that a 95% confidence interval,
as typically computed, contains the parameter in question.

Confidence intervals can be computed for various
parameters, not just the mean. For example, later in this chapter
you will see how to compute a confidence interval for ρ, the
population value of Pearson's r, based on sample data.