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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 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

  1. Two-sided confidence interval for \(\sigma\) $$ \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{1-\alpha/2, N-1} }} \le \sigma \le \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{\alpha/2, N-1} }} \, ,$$

  2. Lower one-sided confidence interval for \(\sigma\) $$ \sigma \ge \frac{s\sqrt{N-1}}{\sqrt{ \chi^2_{1-\alpha, N-1} }} \, ,$$

  3. 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.
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