7.2.3.1. Confidence interval approach

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 $100(1-\alpha)$ % confidence intervals that correspond to the tests of hypothesis on the previous page are given by

Two-sided confidence interval for $\sigma$ $$ \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{1-\alpha/2, N-1} }} \le \sigma \le \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{\alpha/2, N-1} }} \, ,$$
Lower one-sided confidence interval for $\sigma$ $$ \sigma \ge \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{1-\alpha, N-1} }} \, ,$$
Upper one-sided confidence interval for $\sigma$ $$ 0 \le \sigma \le \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{\alpha, N-1} }} \, .$$

For case (1), $\chi_{\alpha/2}^2$ is the $\alpha/2$ 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 $\alpha$ 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, $\sigma_0$, 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 clear-cut 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.