Statistical Methods

Unit 6 - Small samples Tests (n <30)

6.1.3. Case study 3
CASE 3: Test of difference between two means of paired observations (paired t – test)
When the two samples of equal size are drawn from two normal populations and these samples are not independent, then the paired t-test is used. Dependent samples arise, for instance, in experiments when an individual is tested first under one condition and then under another condition, so that there will be two observations for the same individual. Let n be the size of each of the two samples and d1, d2...................dn the difference between the corresponding members of the sample. Let denote the mean of differences and s the standard deviation of these differences.

(iii) Statistical decision
If > the table value at the specified significance, reject Ho at the level.

Example 3 : The following table gives the obtained by 9 students in two tests, one held beginning of a year and the other at the year after intensive coaching. Do the data in the students have benefited by coaching?

 

Student	1	2	3	4	5	6	7	8	9
Test 1	55	60	65	75	49	25	35	18	61
Test 2	63	70	70	81	54	29	32	21	70

Answer:
(i) Hypotheses


(ii) Test statistic

	1	2	3	4	5	6	7	8	9	Total
Test 1 (X)	55	60	65	75	49	25	35	18	61
Test 2 (Y)	63	70	70	81	54	29	32	21	70
d= Y-X	8	10	5	6	5	4	-3	3	9	47
d2	64	100	25	36	25	16	9	9	81	365

(iii) Statistical decision
The table value of t with 8 degrees of freedom is 2.306 at 5% level of significance and 3.355 at 1% level of significance.
Since > the table value of t at both 5% and 1% level of significance, H_o is rejected. In other words it is concluded that the coaching has benefited the students.