NIMCET 2013 — Mathematics PYQ
NIMCET | Mathematics | 2013If X1 and X2 are the means of two distributions such that \overline{X}_1 < \overline{X}_2, and X is the mean of the combined distribution, then:
Choose the correct answer:
- A.
\overline{X} < \overline{X}_1
- B.
\overline{X} > \overline{X}_2
- C.
X=2X1+X2
\overline{X}_1 < \overline{X} < \overline{X}_2
Explanation
Solution
Combined Mean Formula:
Proof:
Since n_1, n_2 > 0 and \overline{X}_1 < \overline{X}_2:
-
X−X1=n1+n2n1X1+n2X2−X1
X−X1=n1+n2n1X1+n2X2−n1X1−n2X1
\overline{X} - \overline{X}_1 = \frac{n_2(\overline{X}_2 - \overline{X}_1)}{n_1 + n_2} > 0 \implies \overline{X} > \overline{X}_1
-
X2−X=X2−n1+n2n1X1+n2X2
X2−X=n1+n2n1X2+n2X2−n1X1−n2X2
\overline{X}_2 - \overline{X} = \frac{n_1(\overline{X}_2 - \overline{X}_1)}{n_1 + n_2} > 0 \implies \overline{X}_2 > \overline{X}
Conclusion:
Explanation
Solution
Combined Mean Formula:
Proof:
Since n_1, n_2 > 0 and \overline{X}_1 < \overline{X}_2:
-
X−X1=n1+n2n1X1+n2X2−X1
X−X1=n1+n2n1X1+n2X2−n1X1−n2X1
\overline{X} - \overline{X}_1 = \frac{n_2(\overline{X}_2 - \overline{X}_1)}{n_1 + n_2} > 0 \implies \overline{X} > \overline{X}_1
-
X2−X=X2−n1+n2n1X1+n2X2
X2−X=n1+n2n1X2+n2X2−n1X1−n2X2
\overline{X}_2 - \overline{X} = \frac{n_1(\overline{X}_2 - \overline{X}_1)}{n_1 + n_2} > 0 \implies \overline{X}_2 > \overline{X}
Conclusion: