Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 44

We find a 95% prediction interval on a future value of propellant shear strength in a motor made from a batch of sustainer propellant that is 10 weeks old. Using (2.45), we find that the prediction interval is

which simplifies to

Therefore, a new motor made from a batch of 10-week-old sustainer propellant could reasonably be expected to have a propellant shear strength between 2048.32 and 2464.32 psi.

Figure 2.5 The 95% confidence and prediction intervals for the propellant data.

Figure 2.5 shows the 95% prediction interval calculated from (2.45) for the rocket propellant regression model. Also shown on this graph is the 95% CI on the mean [that is, E(y|x) from Eq. (2.43). This graph nicely illustrates the point that the prediction interval is wider than the corresponding CI.

We may generalize (2.45) somewhat to find a 100(1 − α) percent prediction interval on the mean of m future observations on the response at x = x₀. Let be the mean of m future observations at x = x₀. A point estimator of is . The 100(1 − α)% prediction interval on is

(2.46)