7.
Product and Process Comparisons
7.2. Comparisons based on data from one process


The testing of H_{0} for a single population mean 
Given a random sample of measurements,
Y_{1}, ..., 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:


Corresponding null hypotheses 
The corresponding null hypotheses that test the true standard
deviation, ,
against the nominal value,
are:


Test statistic 
The basic test statistic is the chisquare statistic
with N  1 degrees of freedom where s is the sample standard deviation; i.e.,


Comparison with critical values 
For a test at significance level
, where
is chosen to be small,
typically 0.01, 0.05 or 0.10, the hypothesis associated with each
case enumerated above is rejected if:
where Χ^{ 2}_{α/2} is the 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. 

Warning  Because the chisquare distribution is a nonnegative, asymmetrical distribution, care must be taken in looking up critical values from tables. For twosided 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, Χ^{ 2}_{0.95, 9} = 16.92. Since the test statistic (17.56) exceeds the critical value (16.92) of the chisquare distribution with 9 degrees of freedom, the manufacturer's claim is rejected. 