t-test (idea)

Cermain's introduction to the t-test is fairly accurate, but I feel lacks some information.

The t-test is a statistical tes twhich allows you to compare two independent samples or to compare a single sample against a theoretical mean. It is a univariate procedure, meaning that you can only compare the values of one variable at a time.

The two statistical hypotheses normally tested are:

• H₀: The mean for sample 1 is equal to the mean for sample two, or μ₁=μs₂
• H₁: The mean for sample 1 is not equal to the mean for sample two, or μ₁ ≠ μ₂. (note that this can be modified to include the unilateral case)

The auxiliary statistic t (the value compared against a theoretical distribution) is calculated as:

t_c = (mean(x₁)-mean(x₂))/s_pd√(1/n₁ +1/n₂)

This value of t is compared against the Student's t distribution with ν=n₁+n₂-2 degrees of freedom. The conditions of the t-test are as follows:

1. The samples be independent
2. The two samples be distributed normally
3. The variances of the two samples are equal

If the two variances are not equal, then a modified version of the test may be applied.

t_mc = (mean(x₁)-mean(x₂))/√(s²_x1 /n₁ +s²_x2/n₂)

Where t_mc is compared with the theoretical Student's t distribution with a modified number of degrees of freedom. There is also a modification of the t-test under the circumstances where the two samples are paired (an example of this situation is where you take a measurement of a subject before and after a manipulation; the samples are not independent because the same subject is measured twice).

t_d = mean(d)/s_mean(d)

where mean(d) and s_d are the mean value and standard deviation of the differences between sample 1 and sample 2 for each subject, and s_mean(d)=s_d/√n and s_d.