When we look at the differences between scores for two groups, we must judge the difference between their means relative to the spread or variability of their scores. The t-test does this.

First, we must ask: Is the observed difference statistically significant?

Second: If the difference is statistically significant, then is it a meaningful or substantive difference? Let's say, for example, that Protestants have a mean of 13.6 years of education and Catholics have a mean of 12.4 years. We find that the observed difference is statistically significant. Does the additional year give Protestants an "advantage" over Catholics?  What does the difference mean?  There is no simple way to answer this question. It depends on the situation and what your argument is (based on theoretical expectations). Substantive differences are different from statistically significant differences.

Statistical Analysis of the t-test

The final combined formula for the t-test is:
 

The t-value will be positive if the mean from Group I is larger than the mean of group II and negative if it is smaller. Once you compute the t-value you look up the t-value in a table of significance which tells us whether the ratio is large enough to say that the difference between the groups is significant.  In other words the difference observed is not likely due to chance or sampling error. As with any test of significance, you need to set the alpha level. In most social research, the "rule of thumb" is to set the alpha level at .05. This means that 5% of the time (five times out of a hundred) you would find a statistically significant difference between the means even if there is none ("chance"). The t-test also requires that we determine the degrees of freedom (df) for the test. In the t-test, the degrees of freedom is the sum of the persons in both groups minus 2. Given the alpha level, the df, and the t-value, you can look the t-value up in a standard table of significance (available as an appendix in the back of most statistics texts) to determine whether the t-value is large enough to be significant. If it is, you can conclude that the difference between the means for the two groups is different. Statistical computer programs routinely print the significance test results, saving you from looking them up in a table.

• observations are independent and not paired (if paired use the paired t Test.).
• observations for each group are a sample from a population that is normally distributed
• variances for the two independent groups must be considered in interpretation of the test statistic.

In the unpaired (unmatched) analysis each unit of analysis (person in this case) is observed or measured only once.
Example: The years of completed education for a group of Protestants and a group of Catholics.

In the paired (matched) situation, each unit of analysis (person in this case) is observed or measured two or more times.
Example: The Math SAT score of a group of people is taken before a course in math and then compared to the Math SAT score taken after the course for the same group of people.   This is an example of the pre-test/post-test experiment.

Note: The t-test, one-way Analysis of Variance (ANOVA) and a form of regression analysis are mathematically equivalent  and would yield identical results.