Question

Last week we were introduced to hypothesis testing using t and z-tests. This week we extend that set of tools to two sample tests. However, here we have a greater variety of tests. For the t-test alone there are three variants based on the independence of the groups and whether we can assume variances the same or variances between the groups as different. The variances issue is easy as we can always assume they are different as the test is robust to accept any variances but the dependent pairs is different. Why is it different and how might you use it?

Answer 1

It has been discussed that, for the t-test, there are three variants based on the independence of the groups and whether we can assume variances the same or variances between the groups as different.
The variances issue is easy as we can always assume they are different as the test is robust to accept any variances but the dependent pairs are different.
paired t-test compares study subjects at 2 different times (paired observations of the same subject). Unpaired t-test (aka Student’s test) compares two different subjects. The paired t-test reduces intersubject variability (because it makes comparisons between the same subject), and thus is theoretically more powerful than the unpaired t-test.

Both tests assume that all data that have been analyzed are normally distributed. Paired t-tests are more comprehensive and compelling than unpaired t-tests because they are done with subjects that have similar characteristics.

the first assumption of t-test is the population from which the sample is to be taken is normally distributed, and sampling technique used to take sample is SRS, and sample size taken should not be very large or very small as well, and one more assumption is that variance should be homogeneous ( large sample and equal number of participants in each group have the effect of minimizing violations of the assumption of homogeneous variance. in addition, the T-test is robust to violations of most of its assumption )

now to differentiate paired and unpaired we will check the condition that the two samples are related if yes then apply paired (dependent) t-test if no then check are the two sample sets of the same size if yes then equal variance (independent) t-test, if no then check are the two sample set have same variance if yes then again equal variance (independent) t-test, if no then unequal variance (independent t-test)

1-sample t-test:- it tests the mean of a single group against known mean

2-sample t-test:- independent sample t-test compare mean for two groups

paired t-test:- to compare the same mean from the same group at different time

Thank you!!

Answer 2

Solution:

The two Independent samples t-test is used, when we have two independent samples from two different populations and we want to test that is there significant difference between the two population means. Now again two independent samples t-test is of two kind. In one kind we assume that population variances of both of the population are same. In second kind the assumption does not hold and two populations have different variances.

Another two samples t-test is dependent samples t-test. Dependent samples t-test is used when we have paired observations. In this testing procedure we test that whether there is a significant difference between mean of two paired groups or not. In this type of test the observations are measured two times for same sampling units before and after some treatment is applied to the sampling units. For example if we want to test that is there any significant difference between the performance of the students before and after a specific teaching method was used. To test this we should have scores of students when the specific teaching method was not used and score of the same students when a specific teaching method was used.

Hence, the basic difference between two independent samples t-test and two dependent samples t-test is that, in independent samples t-test the samples are independent and in the dependent samples t-test the samples are dependent (paired observations).

Now if we have paired observations and we want to test whether, there is any difference between the means of the two paired groups or not, we can use dependent samples t-test. The test statistic used for dependent samples t-test is given as follows:

where, is sample mean of the differences of paired observations, s is sample standard deviation of the differences of paired observations and n is sample size.