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 H₀for a single population mean

Given a random sample of measurements,

, 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 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,

, against the nominal value,

are:

H₀: =
H₀: <
H₀: >

Test statistic

The basic test statistic is the chi-square statistic

with N - 1 degrees of freedom where sis the sample standard deviation; i.e.,

Comparison with critical values

For a test at significance level

, where

is chosen to be small, typically .01, .05 or .10, the hypothesis associated with each case enumerated above is rejected if:

where

is the upper

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,

=0.05, the test statistic,

is compared with the critical value, .

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.