# Confidence Limits

Confidence limits are the lower and upper boundaries of a confidence interval. In our Acme example, the limits were 20 and 24.

## Confidence Level

The confidence level is the probability value attached to a given confidence interval. It can be expressed as a percentage (in our example it is 95%) or a number (0.95).

## Confidence Interval for a Mean

A confidence interval for a mean is a range of values within which the mean (unknown population parameter) may lie. Examples of Confidence Interval for a Mean* A Web master who wishes to estimate her mean daily hits on a certain webpage. * An environmental health and safety officer who wants to estimate the mean monthly spills.

## Confidence Interval for the Difference between Two Means

A confidence interval for the difference between two means specifies a range of values within which the difference between the means of the two populations may lie. Examples of Confidence Interval for the Difference between Two Means* A Web master who wishes to estimate her difference in mean daily visitors between two websites. * An environmental health and safety officer who wants to estimate the difference in mean monthly spills between two production sites.

When we calculate a statistic for example, a mean, a variance, a proportion, or a correlation coefficient, there is no reason to expect that such point estimate would be exactly equal to the true population value, even with increasing sample sizes. There are always sampling inaccuracies, or error. In most Six Sigma projects, there are at least some descriptive statistics calculated from sample data. In truth, it cannot be said that such data are the same as the population’s true mean, variance, or proportion value.

There are many situations in which it is preferable instead to express an interval in which we would expect to find the true population value. This interval is called an interval estimate. A confidence interval is an interval, calculated from the sample data that is very likely to cover the unknown mean, variance, or proportion. For example, after a process improvement a sampling has shown that its yield has improved from 78% to 83%. But, what is the interval in which the population’s yield lies? If the lower end of the interval is 78% or less, you cannot say with any statistical certainty that there has been a significant improvement to the process.

There is an error of estimation, or margin of error, or standard error, between the sample statistic and the population value of that statistic. The confidence interval defines that margin of error. The next page shows a decision tree for selecting which formula to use for each situation. For example, if you are dealing with a sample mean and you do not know the population’s true variance (standard deviation squared) or the sample size is less than 30, than you use the t Distribution confidence interval. Each of these applications will be shown in turn.

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