A confidence interval represents a closed interval where a certain percentage
of the population is likely to lie. For example, a 90% confidence interval
with a lower limit of 
and an upper limit of 
implies that 90% of the population
lies between the values of 
and 
. Out of the remaining 10% of the population,
5% is less than 
and 5% is greater than 
. (For details refer to [19].)
This section discusses confidence intervals used in simple linear regression
analysis.
A 100(
) percent confidence interval on 
is obtained as follows:
(18)
Similarly, a 100(
) percent confidence interval on 
is obtained as:
(19)
A 100(
) percent confidence interval on any
fitted value, 
, is obtained as follows:
(20)
It can be seen that the width of the confidence interval depends on
the value of 
and will be a minimum at 
and will widen as 
increases.
For the data in Table 4.1, assume that a new value of the yield is observed
after the regression model is fit to the data. This new observation is
independent of the observations used to obtain the regression model. If
is the level of the temperature at
which the new observation was taken, then the estimate for this new value
based on the fitted regression model is:
If a confidence interval needs to be obtained on 
, then this interval should include
both the error from the fitted model and the error associated with future
observations. This is because 
represents the estimate for a value
of 
that was not used to obtain the regression
model. The confidence interval on 
is referred to as the prediction
interval
A 100(
) percent prediction interval on a
new observation is obtained as follows:
(21)
Example 4.3
To illustrate the calculation of confidence intervals, the 95% confidence
intervals on the response at 
for the data in Table 4.1 is obtained
in this example. A 95% prediction interval is also obtained assuming that
a new observation for the yield was made at 
.
The fitted value, 
, corresponding to 
is:
The 95% confidence interval 
on the fitted value, 
, is:
The 95% limits on 
are 199.95 and 205.2, respectively.
The estimated value based on the fitted regression model for the new
observation at 
is:
The 95% prediction interval on 
is:
The 95% limits on 
are 189.9 and 207.2, respectively.
In DOE++, confidence and prediction intervals are available using the
Prediction icon in the Control Panel. The prediction interval values calculated
in this example are shown in Figure 4.11 as Low
PI and High PI respectively. The columns labeled Mean Predicted and Standard
Error represent the values of 
and the standard error used in the
calculations.
|
Figure 4.11: Calculation of prediction intervals in DOE++. |
See Also:
Hypothesis Tests in Simple Linear Regression
