TABLE II
Notation and Numerical Calculations for With Trend Over Time
| Notation | Formula | Calculated Value |
|---|---|---|
| Within-lot estimate of slope |  | β̂ = −0.457 |
| Variance of predicted lot means |  | SL2 = 4.197 |
| Mean squared error |  | SE2 = 0.529 |
| Predicted value of Y at t0 | Ŷ = Ȳ* + β̂(t0 − t̄*) | 66.333 at t0 = 12 |
| Estimated variance of Ŷ at t0 |  | 0.648 at t0 = 12 |
| Estimated variance of Y |  | 4.363 |
| Harmonic mean of the number of values in each lota |  | 1.454 |
| Lot degrees of freedom | s = I − 1 | 10 |
| Error degrees of freedom |  | 14 |
| Effective sample size |  | 6.729 at t0 = 12 |
| Upper bound on variance of Y |  | 10.820 |
| Constants used to compute U where χα:s2 and χα:r2 are chi-squared percentiles with area α to the left and s and r degrees of freedom, respectively | C1 = s/χα:s2 − 1 | 1.538 |
| C2 = r/χα:r2 − 1 | 1.131 | |
| Lower bound on one-sided tolerance interval | Equation 6 | L1 = 57.36 |
| Upper bound on one-sided tolerance interval | Equation 7 | U1 = 75.31 |
| Lower and upper on two-sided tolerance interval | Equation 8 | L2 = 57.25, U2 = 75.41 |
| Lower and upper one-sided bounds using HK1 adjustment of two-sided tolerance interval | Equation 8 replacing  with ZP | L1,HK1 = 58.13, U1,HK1 = 74.53 |
↵a This is an adjustment from the value suggested by Park and Burdick (18) but is computationally easier and provides very similar results.