7.
Product and Process Comparisons
7.2. Comparisons based on data from one process 7.2.2. Are the data consistent with the assumed process mean?


Testing using a confidence interval  The hypothesis test results in a "yes" or "no" answer. The null hypothesis is either rejected or not rejected. There is another way of testing a mean and that is by constructing a confidence interval about the true but unknown mean.  
General form of confidence intervals where the standard deviation is unknown 
Tests of hypotheses that can be made from a single sample of data
were discussed on the foregoing page. As
with null hypotheses, confidence intervals can be twosided or
onesided, depending on the question at hand. The general form of
confidence intervals, for the three cases discussed earlier, where
the standard deviation is unknown are:


Confidence level  The confidence intervals are constructed so that the probability of the interval containing the mean is 1  \(\alpha\). Such intervals are referred to as \(100(1\alpha)\) % confidence intervals.  
A 95 % confidence interval for the example 
The corresponding confidence interval for the test of hypothesis
example on the foregoing page is
shown below. A 95 % confidence interval for the population mean of
particle counts per wafer is given by
$$ \begin{eqnarray}
\bar{Y} + \frac{s}{\sqrt{N}} \, (t_{0.025, 9}) \, \le & \mu & \le \, \bar{Y} + \frac{s}{\sqrt{N}} \, (t_{0.975, 9}) \\
& & \\
53.7 + \frac{6.567}{\sqrt{10}} (2.262) \, \le & \mu & \le \, 53.7 + \frac{6.567}{\sqrt{10}}(2.262) \\
& & \\
49.0 \, \le & \mu & \le \, 58.4 \, .
\end{eqnarray} $$


Interpretation  The 95 % confidence interval includes the null hypothesis if, and only if, it would be accepted at the 5 % level. This interval includes the null hypothesis of 50 counts so we cannot reject the hypothesis that the process mean for particle counts is 50. The confidence interval includes all null hypothesis values for the population mean that would be accepted by an hypothesis test at the 5 % significance level. This assumes, of course, a twosided alternative. 