Two-sample t-test is a parametric test used to test the means of two different populations. We have small samples (n<30) taken by the two other distributions.


It required to know:

  • Standard Deviations of the two sample: it can be different or the same in the two sample;
  • Means of the two sample;
  • The two sample size.


Other important assumptions for the use of t-test are:

  • Data need to be independent;
  • The distribution of the data need to be normal, so you need to test it;
  • T-test work badly with outlier; you need to remove them;
  • We need parametric variable (best if they are continuous value).


The hypothesis of this test is like this in case of the two tails test:

  • H0: m1 = m2 that means that the sample1 mean and the sample2 mean are equal;
  • H1m1 != m2 that means that the sample1mean and the sample2 means are different


In case of one tail left you can have for example:

  • H0: m1 = m2 that means that the sample1 mean higher or equal than the sample2 mean;
  • H1m1 < m2 that means that the sample1 mean are lower than the sample2 mean.


In case of one tail right you can have for example:

  • H0: m1 = m2 that means that the sample1 mean higher or equal than the sample2 mean;
  • H1m1 Z m2 that means that the sample1 mean are lower than the sample2 mean.


The way to do this test is similar to the one-sample t-tests. The only difference is the formula for the statistics. You have a different formula for equal o unequal variance.


Two-Sample t-test equal variance

In case of the variance o sample1 and sample2 are equal the formula is the one in image1.

image1 - two sample t-test equal variance
image1 – two sample t-test equal variance

where:

  • m1 and m2 are the means of the two sample;
  • n1 and n2 are the size of the two sample;
  • Sp is the pooled standard deviation that you can calculate with the formula in the image2.
image2 - Pooled standard deviation
image2 – Pooled standard deviation

Remember that, in the T-student table for the critical value, you need to use as a degree of freedom df = n1+n2 -2


Two-Sample t-test unequal variance

In case of the variance o sample1 and sample2 are unequal the formula is the one in image3.

image3 – two sample t-test unequal variance

where:

  • m1 and m2 are the means of the two sample;
  • n1 and n2 are the size of the two sample;
  • S1 and S2 are the standard deviation of the two sample.

In this case the degree of freedom are calculated with the formula in image4.

image4 - degree of freedom for two sample t-test unequal variance
image4 – degree of freedom for two sample t-test unequal variance
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