Many people studying statistics have real confusion with the t Test. It's not an easy concept to grasp. Here is the short description of the t Test that, hopefully, will clarify this statistical method.
The t test is a statistics test generally used to test whether means of populations are different. In the t test, a t value is calculated based upon the difference in the means and variances of the two populations. This t value is compared to a critical t value, which is based upon the required level of certainty (perhaps you need to be at least 95% certain of the outcome) and the degrees of freedom present in the test. If the t value is greater than the critical t value, it could be stated that, within the required degree of certainty, the 2 means are different.
The t value is generally calculated as follows:
t value = (Difference between the group means) / (Variability of the groups)
To sum up the above calculation, if the groups have a high degree of variance (noise), it is harder to tell whether there really is a difference in the means. The higher amount of variance that exists in the groups, the higher will be the denominator of the above fraction. The higher the denominator, the lower will be the t value. The lower the t value, the less likely that the two means are different. The means are considered different if the t value is greater than the critical t value. It is the t value that is compared with a critical t value to determine within a required degree of certainty whether there really is a difference in the means.
The t value's numerator (Difference between group means) and the denominator (variability of the groups) are calculated in difference ways depending on what type of t test is being run.
The t test has a number of variations but the t test is most commonly used to test whether the means of two populations that are normally distributed are equal. This type of t test is referred to as a Student's t test, or sometimes just the student t test. Drawing samples from different populations in this fashion is referred to as an unpaired t test.
A one-sample t test uses one sample drawn from a single population to test whether the population's mean is equal to a specified value. In other words, the one-sample t test is used to test the Null Hypothesis that the population's mean is equal to that specified value. All other types of t tests are variations of the two sample t test.
A paired t test or paired difference t test is use to test whether "before" and "after" measurements taken of a single object are the same. The Null Hypothesis being tested states that there is no difference between "before" and "after." Specifically, the Null Hypothesis states that "after" values - "before" value = 0.
The Student's t test works well if sample sizes are equal even when the populations being compared have unequal variances. A similar t test called Welch's t test works fine for unequal sample sizes taken from populations with unequal variances.
The two populations from which sample are drawn should be normally distributed.
The t test (specifically, the Student's t test) is equivalent to a one-way ANOVA test when it is being applied to test the equality of means of two normally distributed populations.
A one-sample t test uses one sample drawn from a single population to test whether the population's mean is equal to a specified value. In other words, the one-sample t test is used to test the Null Hypothesis that the population's mean is equal to that specified value.
A two-sample t test is used to test whether two separate normally-distributed populations have the same mean. The Student's t test works well if sample sizes are equal but the populations have different variances. Welch's t test works well for unequal sample sizes drawn from populations of unequal variance.
My blog provides a number of articles which show explicit examples of t Tests being performed in Excel.
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