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?


Confidence intervals for the standard deviation 
Confidence intervals for the true standard deviation can be
constructed using the chisquare distribution. The \(100(1\alpha)\) %
confidence intervals that correspond to the
tests of hypothesis on the previous page
are given by
For case (1), \(\chi_{\alpha/2}^2\) is the \(\alpha/2\) critical value from the chisquare distribution with \(N 1\) degrees of freedom and similarly for cases (2) and (3). Critical values can be found in the chisquare table in Chapter 1. 

Choice of risk level \(\alpha\) can change the conclusion  Confidence interval (1) is equivalent to a twosided test for the standard deviation. That is, if the hypothesized or nominal value, \(\sigma_0\), 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 \(\alpha\) can lead to a change in the conclusion. This poses a dilemma. What should \(\alpha\) be? Unfortunately, there is no clearcut answer that will work in all situations. The usual strategy is to set \(\alpha\) small so as to guarantee that the null hypothesis is wrongly rejected in only a small number of cases. The risk, \(\beta\), of failing to reject the null hypothesis when it is false depends on the size of the discrepancy, and also depends on \(\alpha\). The discussion on the next page shows how to choose the sample size so that this risk is kept small for specific discrepancies. 