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IFA Services: Statistics, Two-Sample Student-t Test
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The Student-t Test for Two Samples

Example:
#A = ,
#B = ,

Summary

Characteristics:
This is the standard test for two samples. It is also the yard-stick for
calculating the relative efficiency of other tests. The Student-t test is
the most sensitive test for interval data, but it also requires the most stringent
assumptions.

H0:
Both populations have identical mean values. That is, the difference between
the means is zero.

Assumptions:
Both distributions are Normal distributed with identical variances.
If there is any reason to doubt these assumptions, use another,
distribution-free, test (e.g. the
&B=>
Wilcoxon Test).

Scale:
Interval

Procedure:
Calculate the Mean values (M1, M2) and standard deviations (SD1 and
SD2) of both samples.
Determine SDg = sqrt(((n1-1)*SD1**2 +
(n2-1)*SD2**2)/(n1+n2-2))
The test parameter is t = ( M1 - M2 ) /
( SDg * sqrt( 1 / n1 + 1 / n2 ) ).
The number of Degrees of Freedom = n1 + n2 - 2.

Level of Significance:
The significance levels of t for different Degrees of Freedom
are tabulated.

Approximation:
If the Degrees of Freedom > 30, the distribution of t can
be approximated by a
>
Standard Normal Distribution.

Remarks:
Because of its popularity, this test is very often applied indiscriminately,
when the underlying assumptions are invalid, i.e., when the observations are
not Normal distributed with equal variance. This can lead to
illusory high sensitivities. Although it is the most powerfull test when
the experimental data are indeed Normal distributed with equal
variance, do not hesitate to use a distribution-free
test instead whenever there is some doubt about Normality (the
&B=>
Wilcoxon Test is almost as sensitive).
The name Student-t test is derived from the pen-name of the man who
developed the test. It has nothing to do with the popularity of this test in
introductory courses. WARNING: the level of significance given here is only an approximation,
take care when using it! (use a table if necessary)