A) \[\overline{X}<{{\overline{X}}_{1}}\]
B) \[\overline{X}>{{\overline{X}}_{2}}\]
C) \[\overline{X}=\frac{{{\overline{X}}_{1}}+{{\overline{X}}_{2}}}{2}\]
D) \[{{\overline{X}}_{1}}<\overline{X}<{{\overline{X}}_{2}}\]
Correct Answer: D
Solution :
| Let \[{{n}_{1}}\] and \[{{n}_{2}}\]be the number of observation in two groups having means \[{{\overline{X}}_{1}}\] and \[{{\overline{X}}_{2}}\] respectively. Then |
| \[\overline{X}=\frac{{{n}_{1}}{{\overline{X}}_{1}}+{{n}_{2}}{{\overline{X}}_{2}}}{{{n}_{1}}+{{n}_{2}}}\] |
| Now, \[\overline{X}-{{\overline{X}}_{1}}=\frac{{{n}_{1}}{{\overline{X}}_{1}}+{{n}_{2}}{{\overline{X}}_{2}}}{{{n}_{1}}+{{n}_{2}}}-{{\overline{X}}_{1}}\] |
| \[\frac{{{n}_{2}}({{\overline{X}}_{2}}+{{\overline{X}}_{1}})}{{{n}_{1}}+{{n}_{2}}}>0\,\,[\,\,\because {{\overline{X}}_{2}}>{{\overline{X}}_{1}}]\] | |||
| \[\Rightarrow \overline{X}>{{\overline{X}}_{1}}\] | ??.. (1) | ||
| And, \[\overline{X}-{{\overline{X}}_{2}}=\frac{{{n}_{1}}({{\overline{X}}_{1}}+{{\overline{X}}_{2}})}{{{n}_{1}}+{{n}_{2}}}<0\,\,\,[\,\,\because {{\overline{X}}_{2}}>{{\overline{X}}_{1}}]\] | |||
| \[\Rightarrow \overline{X}<{{\overline{X}}_{1}}\] | ??.. (2) | ||
| From (1) and (2) \[\Rightarrow {{\overline{X}}_{1}}<\overline{X}<{{\overline{X}}_{2}}.\] | |||
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