3
Meaning of a Confidence Level
Think of the confidence level as a success rate. The unknown parameter, perhaps a mean or proportion, is
fixed somewhere on the number line. The confidence interval, on the other hand, is variable. It depends on
which individuals end up in our sample. Most of the time we'll obtain a sample that leads to a confidence
interval that contains the unknown parameter (success!), but sometimes we get unlucky, and our confidence
interval misses the mark (failure!).
Take a look at the diagram to the left. The normal curve
represents the sampling distribution of the sample mean. That
is, it represents all the sample means that we could obtain if we
took all possible samples of a given size. In the center of that
bell curve is the true population mean, μ.
The green lines below the normal curve represent some of the
confidence intervals we could obtain. Most of them include μ
(success!)
. However, you'll notice one confidence interval that
is far off to the right (failure!). That particular sample happened
to have a very high mean, so high that even when the margin of
error was added to both sides, the interval
still
didn't contain µ.
The
confidence level
is the long-term success rate of a confidence interval method in capturing the population
parameter.
o
If you are constructing 95% confidence intervals, for example, then over the long run, 95% of your
confidence intervals will contain the parameter, while 5% will not.
o
In practice we only take
one
sample and construct
one
confidence interval. We do not know if it is a
"success" or "failure".
Interpreting a Confidence Interval
Here is the generic way to interpret a confidence interval in this class:
Do say: "
We are confidence level% confident that the specific mean/proportion for all fill in the blank
population lies between low end of confidence interval and high end of confidence interval units.
"
Example: We are 95% confident that the mean number of times all dogs bark during a day lies between 36.5
and 43.5 barks.
Example: We are 90% confident that the proportion of all orders that receive customer complaints at this
McDonald's lies between 1.4% a
nd 1.8%.
Do not say, "There is a 95%
chance
" that a parameter lies within the interval.
The word "confident" has a specific meaning in statistics, so that is the word we use. The word "chance"
implies that the parameter is moving around, but in reality, it is fixed.