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 that correspond to the tests
of hypothesis on the previous page are given by
- Two-sided confidence interval for
- Lower one-sided confidence interval for
![>= [(SQRT(N-1)*s)/SQRT(Chi-Square(alpha,N-1))]](eqns/sigma1l.gif)
- Upper one-sided confidence interval for
where for case (1)
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.
can change
the conclusion
, is
not contained within these limits, then the hypothesis that the
standard deviation is equal to the nominal value is rejected.
can lead to a change in the conclusion. This poses a dilemma. What
should be?
Unfortunately, there is no clear-cut answer that will work in all
situations. The usual strategy is to set
small so as
to guarantee that the null hypothesis is wrongly rejected
in only a small number of cases. The risk,
, of failing
to reject the null hypothesis when it is false depends on the size
of the discrepancy, and also depends on
. The
discussion on the next page shows how to
choose the sample size so that this risk
is kept small for specific discrepancies.
