7. Product and Process Comparisons
7.2. Comparisons based on data from one process
7.2.3. Are the data consistent with a nominal standard deviation?

7.2.3.1.

Confidence interval approach

Confidence intervals for the standard deviation

Confidence intervals for the true standard deviation can be constructed using the chi-square distribution. The 100(1-)% confidence intervals that correspond to the tests of hypothesis on the previous page are given by

1. Two-sided confidence interval for

   

2. Lower one-sided confidence interval for

   

3. Upper one-sided confidence interval for

   

where for case (1), Χ ²_α/2 is the critical value from the chi-square distribution with N - 1 degrees of freedom and similarly for cases (2) and (3). Critical values can be found in the chi-square table in Chapter 1.

Choice of risk level can change the conclusion

Confidence interval (1) is equivalent to a two-sided test for the standard deviation. That is, if the hypothesized or nominal value, , is not contained within these limits, then the hypothesis that the standard deviation is equal to the nominal value is rejected.

A dilemma of hypothesis testing

A change in can lead to a change in the conclusion. This poses a dilemma. What should be? Unfortunately, there is no clear-cut answer that will work in all situations. The usual strategy is to set small so as to guarantee that the null hypothesis is wrongly rejected in only a small number of cases. The risk, , of failing to reject the null hypothesis when it is false depends on the size of the discrepancy, and also depends on . The discussion on the next page shows how to choose the sample size so that this risk is kept small for specific discrepancies.