## Key Concepts About Using the t-test Statistic

The t-test is used to test the null hypothesis that two population means or
proportions, *θ*_{1} and *θ*_{2,} are equal OR,
equivalently, that the difference between two population means or proportions is
zero. To test this hypothesis, assuming the covariance is small, as is the case
with NHANES data, the following formula is used

#### Equation for t-Test Where Covariance is Small

(1)

where,

_{1} is
an estimate of *θ*_{1} based
on a probability sample,

_{1} is
an estimate of the standard error of _{1},

_{2} is
an estimate of *θ*_{2,}

and _{2} is an
estimate of the standard error of _{2}.

In instances where the t statistic is based on a small number of independent
pieces of information (i.e. a small number of degrees of freedom [<30]), the
statistic given in equation 1 follows a Student's t distribution with mean=0
and unit variance with n degrees of freedom. In the NHANES 1999-2002 sample, the
degrees of freedom depend on the number of first stage units, or PSUs,
containing observations and is defined as the number of PSUs minus the number of
strata. (See Sample Design module for more information.)

The equality of means is usually tested at the .05 level of significance.

**References:**

Cochran, WG. Sampling Techniques. John Wiley & Sons. 1977.

Lohr SL. Sampling: Design and Analysis. Duxbury Press. Pacific Grove 1999.

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