Question

What is the difference between a t test for independent samples and a t test for related (dependent) samples?

Give an example of when you would use each?

Answer 1

INDEPENDENT T TEST: SUPPOSE WE WANT TO TEST IF TWO INDEPENDENT SAMPLES xi(i=1,2,3,...,n1) and yj (j=1,2,3...,n2) of sizes n1 and n2 have been drawn from two normal populations with means and respectively.

Under the null hypothesis H0 that the samples have been drawn from the normal population with means and and under the assumptions that the population variance are equal the statistic

where Sp is pooled stanadrd deviation.

D.F= n1+n2-2

PAIRED T TEST : LET US NOW CONSIDER THE CASE WHEN (i) THE SAMPLE SIZES ARE EQUAL i.e n1=n2=n and (ii) THE TWO SAMPLES ARE NOT INDEPENDENT BUT THE SAMPLES ARE PAIRED TOGETHER (ith) SAMPLE UNIT.

di= xi-yi

Under null hypothesis test statistic is

where dbar= summation di/n

S is standard deviation of difference between two observations.

degrees of freedom= n-1

FROM ABOVE DISCUSSION:

INDEPENDENT T USED WHEN SAMPLES ARE INDEPENDENT ..

POPULATIONS ARE INDEPENDENT AND NORMALLY DISTRIBUTED WITH EQUAL AND UNKNOWN VARIANCES

SAMPLES CAN BE EQUAL OR UNEQUAL.

D.F IS N1+N2-2

EXAMPLE: DIFFERENCE IN AVERAGE GPS OF TWO CLASSES

WHILE PAIRED T TEST USED WHEN SAMPLES ARE PAIRED.

SAMPLE SIZES MUST BE EQUAL

DEPEDENT VARIABLE MUST NOT CONTAIN ANY OUTLIERS.

FOR EXAMPLE: SUPPOSE WE WANT TO TEST EFFICACY OF WEIGHT LOSS DRUG WE WILL USE PAIRED T TEST.