A is rarely equal to the population parameter. Due to this uncertainty, estimations are commonly presented as a confidence interval
. This is a range of values that the actual parameter is expected to fall within with some degree of certainty. A confidence interval is found by adding and subtracting the
to and from the statistic, like the sample
The degree of certainty, or the confidence level
, is usually presented as a . It refers to the reliability of the analysis to produce accurate . For example, if
confidence intervals are produced using different samples of the same size with
intervals are expected to contain the actual mean.
Confidence Level and the Standard Normal Distribution
The confidence level matches the percentage of the under the around the mean limited by the
and , as shown below.
confidence interval, there is a
of observing a value outside this area. Because the distribution is , half of this area will be on each tail of the .
Confidence Interval for the Population Mean
A confidence interval for the population mean can be found by adding and subtracting the maximum error of estimate to and from the sample mean
It is worth noting that increasing the level of confidence results in a wider interval that is more likely to catch the true mean, but it will be less precise because it will cover a greater range of values. This means there is a trade-off between confidence and precision.