t-test

A statistical test for determining if two samples come from the same population. The t-test can be considered a special case of the ANOVA.

Essentially, the t-test works like this: take two samples, and determine the mean and variance for each one. Given this information, you can determine the likelihood that the difference in means is due to random chance. The more likely that it is random, the more likely that the samples came from the same population.

Of course, this can never actually prove, in the strict sense, that two sanples are from the same population, or from different populations; it can just give you probabilities and let you decide what conclusions to make from there. In many sciences, two samples are considered to be from different populations of the probability that they are from the same population is less than five percent.

See null hypothesis.

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.