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 .) 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)
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.
Hypothesis Tests in Simple Linear Regression