Question
Question: If \({\bar{x}}_{1}\) and \({\bar{x}}_{2}\) are the means of two distributions such that \({\bar{x}}_...
If xˉ1 and xˉ2 are the means of two distributions such that xˉ1<xˉ2 and xˉ is the mean of the combined distribution, then
A
xˉ<xˉ1
B
xˉ>xˉ2
C
Xˉ=2Xˉ1+Xˉ2
D
xˉ1<xˉ<xˉ2
Answer
xˉ1<xˉ<xˉ2
Explanation
Solution
Let n1and n2 be the number of observations in two groups having means xˉ1and xˉ2 respectively. Then,
xˉ=n1+n2n1xˉ1+n2xˉ2
Now, xˉ−xˉ1=n1+n2n1xˉ1+n2xˉ2−xˉ1
=n1+n2n2(xˉ2−xˉ1)>0,[∵xˉ2>xˉ1]
⇒ xˉ>xˉ1 .....(i)
and xˉ−xˉ2=n1+n2n(xˉ1−xˉ2)<0, [∵xˉ2>xˉ1]
⇒ xˉ<xˉ2 ......(ii)
From (i) and (ii), xˉ1⥂<xˉ<xˉ2.