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# The Student-t Test for One Sample

Example:

The observation pairs (either a pair each line or the difference itself)

Characteristics:
This is the standard test for matched-pairs. 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:
The mean value of the difference is zero.

Assumptions:
The difference is Normal distributed. If there is any reason to doubt this assumption, use another, distribution-free, test (e.g., > Wilcoxon Matched-Pairs Signed-Ranks Test).

Scale:
Interval

Procedure:
Calculate the Mean value and standard deviation (SD) of the differences, determine the number of sample pairs, N, and Degrees of Freedom (= N-1, in this case).
The test parameter is t = ( Mean / SD ) * sqrt( N ).
If necessary, Mean can be replaced by (Mean - expected value).

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. This can lead to illusory high sensitivities. Although it is the most powerfull test when the experimental data are indeed Normal distributed, do not hesitate to use a distribution-free test instead whenever there is some doubt about Normality (the > Wilcoxon Matched-Pairs Signed-Ranks 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)

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