The formulas for statistical intervals given in this section use the following notation:
Notation |
Definition |
---|---|
number of nonmissing values for a variable |
|
mean of variable |
|
standard deviation of variable |
|
100 |
|
100 |
|
100 |
|
parameter |
|
100 |
|
the numerator and |
|
100 |
The values of the variable are assumed to be independent and normally distributed. The intervals are computed using the degrees of freedom as the divisor for the standard deviation . This divisor corresponds to the default of VARDEF=DF in the PROC CAPABILITY statement. If you specify another value for the VARDEF= option, intervals are not computed.
You select the intervals to be computed with the METHODS= option. The next six sections give computational details for each of the METHODS= options.
This requests an approximate simultaneous prediction interval for future observations. Two-sided intervals are computed using the conservative approximations
One-sided limits are computed using the conservative approximation
Hahn (1970c) states that these approximations are satisfactory except for combinations of small , large
, and large
. Refer also to Hahn (1969, 1970a) and Hahn and Meeker (1991).
This requests a prediction interval for the mean of future observations. Two-sided intervals are computed as
One-sided limits are computed as
This requests an approximate statistical tolerance interval that contains at least proportion of the population. Two-sided intervals are approximated by
where .
Exact one-sided limits are computed as
where .
In some cases (for example, if is large),
is approximated by
where and
.
Hahn (1970b) states that this approximation is "poor for very small , especially for large
and large
, and is not advised for
." Refer also to Hahn and Meeker (1991).
This requests a confidence interval for the population mean. Two-sided intervals are computed as
One-sided limits are computed as