Q: Does a null hypothesis always state that there is no difference? Could there be a null hypothesis that claims there is?
In the U.S. legal system, the null hypothesis is that the accused is assumed innocent until proven guilty. In another legal system, there might exist the possibility that the accused is assumed guilty until proven innocent. In our system, a type 1 error would be to find an innocent man guilty. What would be considered a type 1 error if the null hypothesis was assumed guilt?
A: Sir Ronald Fisher developed this basic principle more than 90 years ago. As you have correctly stated above, the process is assumed innocent until proven guilty. You must have evidence beyond reasonable doubt. An alpha error (type 1) is calling an innocent person guilty. Failure to prove guilt when a person really did commit a crime is a Beta error (type 2).
What can null hypothesis tell us? Does the confidence interval include zero (or innocence in the court example)? Instead of asking, “can you assume guilt and prove innocence?” — turn the question around and ask “does the confidence interval include some value that is guilty?”
For example, let’s say a process has an unknown mean and standard deviation, but it has customer specifications from 8-12 millimeters. Your sample measures 14 millimeters. Clearly, your sample is guilty by customer specifications. We need to prove beyond reasonable doubt that the confidence interval of the process, at some risk level (alpha), does not include guilty material. This is done by measuring the process for control. If it is in control and not meeting customer specifications, either move the distribution, reduce the variation (through Design of Experiments, or other methods), or through some combination of both.
If the new confidence interval does not include guilt, the argument would be that you have proven, beyond reasonable doubt, that the confidence interval does not include the out-of-spec material. Under this circumstance, a type 1 error (alpha error) would be a process mean less than the upper specification, but the confidence interval included the specification.
For further reading on this material, refer to the following text:
Testing Statistical Hypothesis, E. L. Lehmann and Joseph Romano, January 2005.
ASQ Certified Six Sigma Master Black Belt
President, William Hooper Consulting Inc.