If ¯¯¯x1 and ¯¯¯x2 are the means of two distributions such that ¯¯¯x1<¯¯¯x2 and ¯¯¯x is the mean of the combined distribution, then
A
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B
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C
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D
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Solution
The correct option is B Let n1 and n2 be the number of observations in two groups having means ¯¯¯¯¯x1 and ¯¯¯¯¯x2 Respectively. Then,¯¯¯x=n1¯¯¯x1+n2¯¯¯x2n1+n2 Now,¯¯¯x−¯¯¯x1=n1¯¯¯x1+n2¯¯¯x2n1+n2−¯¯¯x1 =n2(¯¯¯x2−¯¯¯x1)n1+n2=0,[∵¯¯¯x2>¯¯¯x1] ⇒¯¯¯x>¯¯¯x1.....(i) And¯¯¯x>¯¯¯x2=n(¯¯¯x1−¯¯¯x2)n1+n2<0,[∵¯¯¯x2>¯¯¯x1] ⇒¯¯¯x>¯¯¯x2....(ii) From(i)and(ii)¯¯¯x1<¯¯¯x<¯¯¯x2.