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If overline (x(1)) " and " overline (x(2...

If `overline (x_(1)) " and " overline (x_(2))` are the means of two distributions such that `overline (x_(1)) lt overline (x_(2)) " and " overline (x)` is the mean of the combined distriubtion, then

A

`overline (x) lt overline (x_(1))`

B

`overline (x) gt overline (x_(2))`

C

`overline (x)=(overline (x_(1))+overline (x_(2)))/(2)`

D

`overline (x_(1)) lt overline (x) lt overline (x_(2))`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `n_(1) " and" n_(2)` be the number of observations in two groups having means `overline(x_(1)) " and" overline (x_(2))`, respectively. Then,
`overline (x)=(n_(1)overline(x_(1))+n_(2)overline(x_(2)))/(n_(1)+n_(2))`
Now, `overline (x)-overline(x_(1))=(n_(1)overline(x_(1))+n_(2)overline(x_(2)))/(n_(1)-n_(2))-overline(x_(1))`
` =(n_(2)(overline(x_(2))-overline(x_(1))))/(n_(1)+n_(2)) gt 0 " " [because overline(x_(2))gt overline(x_(1))]`
`implies overline(x) gt overline(x_(1))`,
and `overline(x)-overline(x_(2))=(n(overline(x_(1))-overline(x_(2))))/(n_(1)+n_(2)) lt 0 " " [ because overline(x_(2)) gt overline(x_(1))]`
`implies overline (x) lt overline (x_(2))`
From (1) and (2), `overline(x_(1)) lt overline(x) lt overline(x_(2))`.
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