8.19 Distribution of 't' for comparison of two samples Means Independent samples
Let x1i
( i = 1, 2, 3, ....., n_{1}
) and x2i
( i = 1, 2, 3, ....., n_{2}
) be two random independent samples drawn from two normal populations
with mean m_{1}
and m_{2}
respectively but with same variance s^{2}.
Let
and
be sample means and let
Then the static t is given by
is called the
pooled estimate of the population variance.
Example Two types of drugs were used
on 5 and 7 patients for reducing their weights in Jerry’s 'slimbeauty'
health club. Drug A was allopathic and drug B was Herbal. The decrease
in the weight after using drugs for six months was as follows :
Drug A : 10 12 13 11 14
Drug B : 8 9 12 14 15 10 9
Is there a significant difference in the efficiency of the two drug ? If not which drug should you buy ?
Solution:
Let the null hypothesis H_{o} : m_{1}
= m_{2}
or H_{o} : m_{1}
 m_{2}
= 0.
Alternative
hypothesis H_{a}
: m_{1}
¹
m_{2}
or H_{a}
: m_{1}
 m_{2}
¹ 0
Thus the null hypothesis is accepted. Hence
there is no significance in the efficiency of the two drugs. Since
drug B is Herbal and there is no difference in efficiency between
the two with no side effects, we should buy the Herbal drug.
