t test for Independent Groups
Purpose: The test is used to infer whether two populations have different means on a particular outcome variable. The test examines whether the observed difference between sample means from two different groups is more than would be expected due to sampling error alone.
Examples: Is there a difference between a treatment and a control group on some outcome measure? Are male students scoring differently than female students on a particular test?
Test Statistic: The numerator of the t statistic is your best estimate of the average difference between the two groups. The denominator is your best estimate about how much sampling error there is likely to be in such an estimate.
Critical Value: The critical value is the t statistic with n1 + n2 - 2 degrees of freedom which cuts off the regionfor a one tailed test or the regionfor a two tailed test; or the F statistic with 1 degree of freedom in the numerator and n1 + n2 - 2 degrees of freedom in the denominator, which cuts off the regionof the F distribution
Assumptions:Normality, that the scores on the dependent variable in each of the two populations are distributed normally; Homogeneity of variance, that the scores on the dependent variable in each of the two populations are spread out around the mean to the same extent; Independence of observations, that the score for one observation does not in any way influence the score on another observation; the subjects are randomly selected.
Null Hypothesis:1 = 2 or 1 - 2 = 0
Directional Alternative Hypothesis:12 or 1 - 2 > 0
Non-directional Alternative Hypothesis:1 =/= 2 or 1 - 2 =/= 0
where:
1 = population mean for group one
2 = population mean for group two
Special Considerations: There are two versions of this test, one for use when pooling the variance estimates across the two groups and one for use when separate variance estimates are justified due to heterogeneity of variance.
The same hypotheses can be tested with a confidence interval approach by placing a (1-) percent confidence interval using the t statistic around the difference between the means and then observing if the interval captures 0.
t test for Dependent Means
Purpose: The test is used to infer whether the same population has two different means on a particular outcome variable when it is measured under two different conditions. The test examines whether the observed difference between sample means obtained from the same group over two different conditions is more than would be expected due to sampling error alone.
Examples: Does a group exposed to a particular treatment score higher on a posttest than they did on a pretest? Do the test scores of my students improve from the beginning of the year to the end of the first semester?
Test Statistic: The numerator of the t statistic is your best estimate of the average difference between condition one and condition two. The denominator is your best estimate about how much sampling error there is likely to be in such an estimate.
Critical Value: The critical value is the t statistic with n - 1 degrees of freedom which cuts off the regionfor a one tailed test or the regionfor a two tailed test; or the F statistic with 1 degree of freedom in the numerator and n - 1 degrees of freedom in the denominator which cuts off the region of the F distribution
Assumptions: Normality, that the scores on the dependent variable in each of the two populations are distributed normally; Homogeneity of variance, that the scores on the dependent variable in each of the two populations are spread out around the mean to the same extent; Independence of observations, that the score for one observation does not in any way influence the score on another observation; the subjects are randomly sampled.
Null Hypothesis:d = 0 or 1 - 2 = 0
Directional Alternative Hypothesis:d > 0 or 1 - 2 > 0
Non-directional Alternative Hypothesis:d =/= 0 or 1 - 2 =/= 0
where:
d = the average difference between condition one and two in
the population.
Special Considerations: There are four conditions when this test is useful: 1.) testing the same group at two times, say pretest and posttest, 2.) testing the same group under two conditions, say with tutoring and without, 3.) when two separate groups are matched so that they act as if they are the same group under two conditions, and 4.) when you are analyzing naturally occurring pairs such as husband and wife, twins, etc.
The same hypotheses can be tested with a confidence interval approach by placing a (1-) percent confidence interval using the t statistic around the average difference between condition one and condition two and then observing if the interval captures 0.
One Sample t test for One Mean
Purpose: The test is used to infer whether one population has a mean on a particular outcome variable that is different from some known population value. The test examines whether the observed difference between one sample mean and a population mean of interest is more than would be expected due to sampling error alone.
Examples: Does the new registration system take less time on average to register a student than the old system? Are my students scoring above the national average?
Test Statistic: The numerator of the t statistic is your best estimate of the average difference between the group of interest and the comparison population value. The denominator is your best estimate about how much sampling error there is likely to be in such an estimate. It is also, in the case of the one sample test, an estimate of the population standard error of the mean.
Critical Value: The critical value is the t statistic with n - 2 degrees of freedom which cuts off the regionfor a one tailed test or the regionfor a two tailed test
Assumptions: Normality, that the scores on the dependent variable in the population are distributed normally; Independence of observations, that the score for one observation does not in any way influence the score on another observation; that the population variance is not known but is estimated from sample data; the subjects are randomly sampled.
Null Hypothesis: = 0 or - 0 = 0
Directional Alternative Hypothesis:0 or - 0 > 0
Non-directional Alternative Hypothesis: =/= 0 or - 0 =/= 0
where:
= the population mean for the group of interest
0 = the population mean against which the comparison is made.
Special Considerations: The same hypotheses can be tested with a confidence interval approach by placing a (1-) percent confidence interval using the t statistic around the difference between the group mean and the population mean and then observing if the interval captures 0.