Basic Analysis and Graphing - SAS

72 Performing Univariate Analysis Chapter 2
Statistical Details for the Distribution Platform
Statistical Details for Prediction Intervals
The formulas that JMP uses for computing prediction intervals are as follows:
• For m future observations:
y
for
˜m y 1
[ , m˜ ] = X±
t ( 1 – α ⁄ 2m;
n – 1)
× 1 + -- × s m ≥ 1
n
• For the mean of m future observations:
[ Y l
, Y u
] X t ( 1 – α ⁄ 2,
n – 1)
---
1 1
= ±
× + -- × s for m ≥ 1 .
m n
• For the standard deviation of m future observations:
[ s l
, s u
] = s × --------------------------------------------------------
1
, s × F for
F ( 1 – α ⁄ 2;(
m – 1,
n – 1)
) m ≥ 2
( 1 – α ⁄ 2;(
n – 1,
m – 1)
)
where m = number of future observations, and n = number of points in current analysis sample.
• The one-sided intervals are formed by using 1-α in the quantile functions.
For references, see Hahn and Meeker (1991), pages 61-64.
Statistical Details for Tolerance Intervals
This section contains statistical details for one-sided and two-sided tolerance intervals.
One-Sided Interval
The one-sided interval is computed as follows:
Upper Limit = x+
g's
Lower Limit = x–
g's
where
g' = t( 1 – α, n – 1,
Φ 1 ( p) ⋅ n) ⁄ n from Table 1 of Odeh and Owen (1980).
t is the quantile from the non-central t-distribution, and
Φ – 1
is the standard normal quantile.
Two-Sided Interval
The two-sided interval is computed as follows:
[ T˜ p , T˜ L p ] = [ x – g U ( 1 – α ⁄ 2 ; pL , n) s , x+ g ( 1 – α ⁄ 2 ; p U
, n)
s ]
where