Elementary Statistics and Computer Application

Equality of two means (Dependant samples)

Paired t test

• In the t-test for difference between two means, the two samples were independent of each other. Let us now take particular situations where the samples are not independent


• In agricultural experiments it may not be possible to get required number of homogeneous experimental units. For example, required number of plots which are similar in all; characteristics may not be available. In such cases each plot may be divided into two equal parts and one treatment is applied to one part and second treatment to another part of the plot. The results of the experiment will result in two correlated samples. In some other situations two observations may be taken on the same experimental unit. For example, the soil properties before and after the application of industrial effluents may be observed on number of plots. This will result in paired observation. In such situations we apply paired t test.


• Suppose the observation before treatment is denoted by x and the observation after treatment is denoted by y. for each experimental unit we get a pair of observation(x,y). In case of n experimental units we get n pairs of observations : (x1,y1), (x2,y2)…(xn,yn). In order to apply the paired t test we find out the differences (x1- y1), (x2-y2),..,(xn-yn) and denote them as d1, d2,…,dn. Now d1, d2…form a sample . we apply the t test procedure for one sample (i.e)






• the mean may be positive or negative. Hence we take the absolute value as . The test statistic t follows a t distribution with (n-1) d.f.
Example 5

• In an experiment the plots where divided into two equal parts. One part received soil treatment A and the second part received soil treatment B. each plot was planted with sorghum. The sorghum yield (kg/plot) was absorbed. The results are given below. Test the effectiveness of soil treatments on sorghum yield.
Soil treatment A	49	53	51	52	47	50	52	53
Soil treatment B	52	55	52	53	50	54	54	53
Solution

• H0 : µ1 = µ2 , there is no significant difference between the effects of the two soil treatments
• H1 : µ1 #µ2, there is significant difference between the effects of the two soil treatments
• Level of significance = 5%
test statistic



x	y	d=x-y	d2
49	52	-3	9
53	55	-2	4
51	52	-1	1
51	52	-1	1
47	50	-3	16
50	54	-4	16
52	54	-2	4
53	53	0	0
Total		-16	44








• Table value of t for 7 d.f. at 5% l.o.s is 2.365
Inference
t>ttab

• We reject the null hypothesis H0. We conclude that the is significant difference between the two soil treatments between A and B Soil treatment B increases the yield of sorghum significantly,