# The TTEST Procedure

#### One-Sample Design

Define the following notation:

##### Normal Data (DIST=NORMAL)

The mean estimate , standard deviation estimate s, and standard error are computed as follows:

The confidence interval for the mean is

The t value for the test is computed as

The p-value of the test is computed as

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

The acceptance region can be algebraically manipulated to give the following confidence interval for :

Taking the square root of each side yields the CI=EQUAL confidence interval for :

The other confidence interval for the standard deviation (CI=UMPU) is derived from the uniformly most powerful unbiased test of (Lehmann, 1986). This test has acceptance region

where the critical values and satisfy

and

where is the PDF of the chi-square distribution with degrees of freedom. This acceptance region can be algebraically manipulated to arrive at

where and 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 and for and ; the resulting and sum to . Hence, a confidence interval for is given by

Taking the square root of each side yields the CI=UMPU confidence interval for :

##### 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

First, a DIST=NORMAL analysis is performed on with the null value , producing the mean estimate , the standard deviation estimate , a t value, and a p-value. The geometric mean estimate and the CV estimate of the original lognormal data are computed as follows:

The t value and p-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 is derived from and is derived from .