7.2.3. Are the data consistent with a nominal standard deviation?

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?

The testing of $H_0$ for a single population mean

Given a random sample of measurements, $Y_1, \, \ldots, \, Y_N$, there are three types of questions regarding the true standard deviation of the population that can be addressed with the sample data. They are:

Does the true standard deviation agree with a nominal value?
Is the true standard deviation of the population less than or equal to a nominal value?
Is the true stanard deviation of the population at least as large as a nominal value?

Corresponding null hypotheses

The corresponding null hypotheses that test the true standard deviation, $\sigma$, against the nominal value, $\sigma_0$, are:

$H_0: \,\, \sigma = \sigma_0$
$H_0: \,\, \sigma \le \sigma_0$
$H_0: \,\, \sigma \ge \sigma_0$

Test statistic

The basic test statistic is the chi-square statistic $$ \chi^2 = \frac{(N-1)s^2}{\sigma_0^2} \, , $$ with $N-1$ degrees of freedom where $s$ is the sample standard deviation; i.e., $$ s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N \left( Y_i - \bar{Y} \right)^2} \, . $$

Comparison with critical values

For a test at significance level $\alpha$, where $\alpha$ is chosen to be small, typically 0.01, 0.05 or 0.10, the hypothesis associated with each case enumerated above is rejected if: $$ \begin{eqnarray} \mbox{1. } & \chi^2 \ge \chi^2_{1-\alpha/2} \,\, \mbox{ or } \,\, \chi^2 \le \chi^2_{\alpha/2} \\ & \\ \mbox{2. } & \chi^2 \ge \chi^2_{1-\alpha} \\ & \\ \mbox{3. } & \chi^2 \le \chi^2_{\alpha} \, , \\ \end{eqnarray} $$ where $\chi^2_{\alpha/2}$ 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.

Warning

Because the chi-square distribution is a non-negative, asymmetrical distribution, care must be taken in looking up critical values from tables. For two-sided tests, critical values are required for both tails of the distribution.

Example

A supplier of 100 ohm^.cm silicon wafers claims that his fabrication process can produce wafers with sufficient consistency so that the standard deviation of resistivity for the lot does not exceed 10 ohm^.cm. A sample of $N$ = 10 wafers taken from the lot has a standard deviation of 13.97 ohm.cm. Is the suppliers claim reasonable? This question falls under null hypothesis (2) above. For a test at significance level, $\alpha$ = 0.05, the test statistic, $$ \chi^2 = \frac{(N-1)s^2}{\sigma_0^2} = \frac{9(13.97)^2}{100} = 17.56\, , $$ is compared with the critical value, $\chi_{0.95, \, 9}^2$ = 16.92.

Since the test statistic (17.56) exceeds the critical value (16.92) of the chi-square distribution with 9 degrees of freedom, the manufacturer's claim is rejected.