Six Sigma Tutorial -> Z Confidence Intervals for Mean

Six Sigma Z Confidence Intervals for Means

Z Confidence Interval for Means applies to a mean from a normal distribution of variable data. Use the normal distribution for the confidence interval for a mean if the sample size n is relatively large (≥ 30), and σ is known.

The confidence interval (C.I.) includes the shaded area under the curve in between the critical values, excluding the tail areas (the α risk). The entire curve represents the most likely distribution of population means, given the sample's size, mean, and the population's standard deviation.

Here we are making an assumption that the underlying data we are working with is distributed like the bell curve shown.

The most common confidence interval used in industry is probably the 95% confidence interval. If we were to use its formula on many sets of data from the population, then 95% of the intervals would contain the unknown population mean that we are trying to estimate. And 5% of the intervals would not contain the population mean. 2.5% of the time, the interval would be low, and 2.5% of the time, the interval would be too high.

The probability is 95% that the interval contains the population parameter. The 95% value is the confidence coefficient, or the degree of confidence. The end points of the interval are called the confidence limits. In the graphic on the previous page, the endpoints are defined by

Example - Z Confidence Interval for Means
 

Calculate a 95% C.I. on the mean for a sample (n = 35) with an x-bar of 15.6" and a known σ of 2.3 "

This interval represents the most likely distribution of population means, given the sample's size, mean, and the population's standard deviation. 95% of the time, the population's mean will fall in this interval.