Because it is desired to center at the GMP average in this example, the center of the interval is 89.5. This center estimate
involves only three GMP lots, so c = 3. The value of k
_{2 }using equation (1) with c = 3, r = 72, α = 0.05, and p = 0.99 is
The computed tolerance interval with the center of 89.5 and RMSE = 1.64 is from 83.8 (lower limit) to 95.2 (upper limit).
Note that this interval is much tighter than the previously computed interval from 67.4 to 112. This is largely because k
_{2} has decreased from 13.1 to 3.45. By making use of all the available data, a more meaningful interval has been obtained.
As noted previously, it is often expected that OPs will vary around the setpoint value. Using the simulator tool in JMP 6.0,
one can model this behavior and use it to construct a tolerance interval. To demonstrate this process, assume that in our
example we are confident that OP1 will be fixed at setpoint, but that OP2 and OP3 will randomly drift around their setpoints,
but within their respective ORs, in accordance to some specified probability distribution. The following algorithm can be
used to simulate a tolerance interval based on these assumptions and the assumed regression model:
1. Simulate values of OP2 and OP3 from appropriate probability distributions.
2. Compute the predicted value of the PP using the fitted regression model for the simulated values of OP2 and OP3 and the
fixed value of OP1.
3. Add a suitably chosen error term to account for uncertainty in the model fit.
4. Perform steps 1–3 a large number of times, say 100,000 times. The resulting set of 100,000 observations is an empirically
derived set of PP values. Take as the tolerance interval the range that includes the middle 99% of these values. (This is
the range bounded by the 0.5 and 99.5 percentiles.)
Figure 4

Figure 4 presents the JMP simulator panel with the input values for this simulation. The behavior of OP2 is modeled with a
uniform distribution and the selected distribution for OP3 is the triangular distribution. In this case, it was expected that
OP3 would generally move below the setpoint value of 45, and the triangular distribution describes this type of movement.
JMP has a variety of distributions that can be selected to describe movement of the OP.
Figure 5

The simulated empirical distribution of 100,000 PP values is shown in Figure 5. The simulated distributions of OP2 and OP3
are shown in Figure 6.
