Confidence Intervals, although valuable, are not used as often as they should be. Herein, two very important questions shall be answered. Why are point estimates useless for making decisions? And, what is the best confidence level?
Point estimate when found for 5 measurements 57, 55, 56, 55, and 57 will come up as 56. This is not a true value, since the population mean can never be known with certainty. But, this does give an estimate. Rather than a point estimate, the confidence intervals seem to makes more sense which says that “I am 95 percent confident that μ is between 54.76 and 57.24”.
If we assume the mean is 56 and standard deviation is 1, for a certain example, 95% of the times this value is correct.
In another example, where the capability of the population needs to be determined, the sample size matters, to be able to judge if the point estimate is correct or not. A statement such as, “I am 97 percent confident that Cpk is at least 1.50”, might sound plausible. For Cpk, this might be more appropriately called the ‘lower confidence bound’.
Since confidence interval takes sample size and assumption into consideration, they are more dependable when making a decision.
Now, what can be considered as the best confidence level? It depends on the business decision that needs to be taken. In the first example, if 56 is inside the interval, the process is working fine, but if it is outside, the process needs to be adjusted. Again, if Cpk > 1.50, the process is accepted, but if 1.50 comes to be more than the lower confidence bound, the process is disallowed.
There is a possibility of two kinds of errors using confidence intervals. Type I error (false alarms/ producer’s risk) is when the decision value falls outside the interval when it should have been inside. And Type II error (missed detection/ consumer’s risk) is just the opposite. Here the decision value falls inside the interval when it should have been outside.
The likelihood of Type I error, ‘α’, is always measured to be 1 minus the confidence level. The odds of a Type II error, ‘β’, lies on the sample size and also on the position of the true population value. The probability of α and β can be adjusted depending on the severity of the issue. While increasing the sample size or even by decreasing the confidence level, β can be decreased and α can be decreased by increasing the confidence level.
Therefore, depending on the cases, confidence interval is a much more adaptable, and reliable measurement criteria for a Six Sigma process. Rather than a point estimate, they effect decision making more efficiently.