Interpretation of a confidence interval
In practice we only have a single sample (and a single confidence interval).
We do not know whether or not this interval actually includes the unknown population parameter. However knowing that confidence intervals obtained in this way will usually include it is very helpful. In practice,...
Being right most of the time is the best one can hope for, since there is always the possibility of being misled by an unlucky sample.
95% confidence level
We use the scenario from the previous page (estimating a population mean,
µ,
from a sample mean, ,
when the population standard deviation, σ,
is known) to illustrate some general properties of confidence intervals.
Since the sample mean, ,
varies from sample to sample, so does the 95% confidence interval that is
obtained from the sample — it too is a random quantity.
Most samples will lead to confidence intervals that include µ, but some will not. However the proportion of 95% confidence intervals (obtained by repeated sampling from the same population) that include µ is 0.95.
Simulation to demonstrate the confidence level
The diagram below shows a random sample from a normal population with µ = 12 and σ = 2.
The 95% confidence interval is narrow when drawn on the same scale as the sample data, so click the checkbox Expanded summary scale to draw it on a more appropriate scale.
Click Take sample a few times to examine the interval estimates that would arise from different samples from this population. Click the checkbox Accumulate then take about 100 more samples. (Hold down the Take sample button.) You will probably observe that some intervals (drawn in red) do not include µ = 12.
Click on intervals on the right to see the random samples from which they were obtained. Observe that the intervals that do not include 12.0 come from samples whose individual values are not particularly extreme.
After about 200 samples, you should observe that approximately 95% of the interval estimates include the value 12.0.
In practice, only a single sample would be available. We cannot tell whether it is one of the 'lucky 95%' whose confidence intervals include µ, but we are 95% confident that it is.