The response at each treatment of a single factor experiment can be
assumed to be a normal population with a mean of and variance of provided that the error terms can
be assumed to be normally distributed. A point estimator of is the average response at each treatment,
. Since this is a sample average, the
associated variance is , where is the number of replicates at the
th treatment. Therefore, the confidence
interval on is based on the distribution. Recall from Chapter
3 (inference on population mean when variance is unknown) that: [Note]
has a distribution with degrees of freedom
. Therefore, a 100() percent confidence interval on the
th treatment mean, , is:(12)For example, for the first treatment
of the lathe speed we have:
In DOE++, this value
is displayed as the Estimated Mean for the first level, as shown in the
Data Summary Table in Figure 6.3. The value displayed
as the standard deviation for this level is simply the sample standard
deviation calculated using the observations corresponding to this level.
[Note]
The 90% confidence interval for this treatment is:
Figure 6.3: Data Summary table for the single factor experiment
in Table 6.1.
The confidence interval on the difference in two treatment means, , is used to compare two levels of
the factor at a given significance. If the confidence interval does not
include the value of zero, it is concluded that the two levels of the
factor are significantly different. The point estimator of is . The variance for is:For balanced designs all . Therefore:The standard deviation for can be obtained by taking the square
root of and is referred to as the pooled
standard error:The statistic for the difference is:Then a 100(1-) percent confidence interval on the
difference in two treatment means, , is:(13)For example, an estimate of the
difference in the first and second treatment means of the lathe speed,
, is:The pooled standard error for this
difference is:To test , the statistic is:In DOE++, this value is displayed
in the Mean Comparisons table under the column T Value as shown in Figure
6.4. The 90% confidence interval on this difference
is:Hence the 90% limits on () are and , respectively. These values are displayed
under the Low CI and High CI columns in Figure 6.4.
Since the confidence interval for this pair of means does not included
zero, it can be concluded that these means are significantly different
at 90% confidence. This conclusion can also be arrived at using the value noting that the hypothesis is
two-sided. The value corresponding to the statistic
, based on the distribution with 9 degrees of freedom
is:Since value < 0.1, the means are significantly
different at 90% confidence. Bounds on the difference between other treatment
pairs can be obtained in a similar manner and it is concluded that all
treatments are significantly different.
Figure 6.4: Mean Comparisons table for the data in Table
6.1.
and variance of 
provided that the error terms can
be assumed to be normally distributed. A point estimator of 
is the average response at each treatment,
. Since this is a sample average, the
associated variance is 
, where 
is the number of replicates at the
th treatment. Therefore, the confidence
interval on 
is based on the 
distribution. Recall from Chapter
3 (inference on population mean when variance is unknown) that: [Note]
has a 
distribution with degrees of freedom
. Therefore, a 100(
) percent confidence interval on the
th treatment mean, 
, is:
(12)For example, for the first treatment
of the lathe speed we have:
(13)
