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The CAPABILITY Procedure

Methods for Computing Statistical Intervals

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

100th percentile of the standard normal distribution

100th percentile of the central distribution with degrees of freedom

100th percentile of the noncentral distribution with noncentrality

 

parameter and degrees of freedom

100th percentile of the F distribution with degrees of freedom in

 

the numerator and degrees of freedom in the denominator

100th percentile of the distribution with degrees of freedom.

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.

METHODS=1

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).

METHODS=2

This requests a prediction interval for the mean of future observations. Two-sided intervals are computed as

One-sided limits are computed as

METHODS=3

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).

METHODS=4

This requests a confidence interval for the population mean. Two-sided intervals are computed as

One-sided limits are computed as

METHODS=5

This requests a prediction interval for the standard deviation of future observations. Two-sided intervals are computed as

One-sided limits are computed as

METHODS=6

This requests a confidence interval for the population standard deviation. Two-sided intervals are computed as

One-sided limits are computed as

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