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If overline (x_(2)) " 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 |
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Answer» `overline (x) LT overline (x_(1))` `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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