A statistic 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 maximum error of estimate $E$ to and from the statistic, like the sample mean $\bar{x}.$ The degree of certainty, or the confidence level, is usually presented as a percent value. It refers to the reliability of the analysis to produce accurate intervals. For example, if $10$ confidence intervals are produced using different samples of the same size with $90\%$ confidence, then $9$ out of $10$ intervals are expected to contain the actual mean.

Confidence Level and the Standard Normal Distribution

The confidence level matches the percentage of the area under the standard normal curve around the mean limited by the $z$ and $\text{-}z$ values, as shown below. For a $99\%$ confidence interval, there is a $1\%$ probability of observing a value outside this area. Because the distribution is symmetric, half of this area will be on each tail of the distribution.

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 $E$ to and from the sample mean $\bar{x}.$

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.