PROC TTEST: One-Sample Design :: SAS/STAT(R) 9.22 User's Guide

The TTEST Procedure

One-Sample Design

Define the following notation:

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

Normal Data (DIST=NORMAL)

The mean estimate $\text{[math]}$ , standard deviation estimate $\text{[math]}$ , and standard error $\text{[math]}$ are computed as follows:

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

The $\text{[math]}$ confidence interval for the mean $\text{[math]}$ is

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

The $\text{[math]}$ value for the test is computed as

$\text{[math]}$

The $\text{[math]}$ -value of the test is computed as

$\text{[math]}$

The equal-tailed confidence interval for the standard deviation (CI=EQUAL) is based on the acceptance region of the test of $\text{[math]}$ that places an equal amount of area ( $\text{[math]}$ ) in each tail of the chi-square distribution:

$\text{[math]}$

The acceptance region can be algebraically manipulated to give the following $\text{[math]}$ confidence interval for $\text{[math]}$ :

$\text{[math]}$

Taking the square root of each side yields the $\text{[math]}$ CI=EQUAL confidence interval for $\text{[math]}$ :

$\text{[math]}$

The other confidence interval for the standard deviation (CI=UMPU) is derived from the uniformly most powerful unbiased test of $\text{[math]}$ (Lehmann 1986). This test has acceptance region

$\text{[math]}$

where the critical values $\text{[math]}$ and $\text{[math]}$ satisfy

$\text{[math]}$

and

$\text{[math]}$

where $\text{[math]}$ is the p.d.f. of the chi-square distribution with $\text{[math]}$ degrees of freedom. This acceptance region can be algebraically manipulated to arrive at

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ solve the preceding two integrals. To find the area in each tail of the chi-square distribution to which these two critical values correspond, solve $\text{[math]}$ and $\text{[math]}$ for $\text{[math]}$ and $\text{[math]}$ ; the resulting $\text{[math]}$ and $\text{[math]}$ sum to $\text{[math]}$ . Hence, a $\text{[math]}$ confidence interval for $\text{[math]}$ is given by

$\text{[math]}$

Taking the square root of each side yields the $\text{[math]}$ CI=UMPU confidence interval for $\text{[math]}$ :

$\text{[math]}$

Lognormal Data (DIST=LOGNORMAL)

The DIST=LOGNORMAL analysis is handled by log-transforming the data and null value, performing a DIST=NORMAL analysis, and then transforming the results back to the original scale. This simple technique is based on the properties of the lognormal distribution as discussed in Johnson, Kotz, and Balakrishnan (1994, Chapter 14).

Taking the natural logarithms of the observation values and the null value, define

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

First a DIST=NORMAL analysis is performed on $\text{[math]}$ in place of $\text{[math]}$ . The geometric mean estimate $\text{[math]}$ and CV estimate $\text{[math]}$ of the original lognormal data are computed as follows:

	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$

The $\text{[math]}$ value and $\text{[math]}$ -value remain the same. The confidence limits for the geometric mean and CV on the original lognormal scale are computed from the confidence limits for the arithmetic mean and standard deviation in the DIST=NORMAL analysis on the log-transformed data, in the same way that $\text{[math]}$ is derived from $\text{[math]}$ and $\text{[math]}$ is derived from $\text{[math]}$ .

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