JEE MAIN - Mathematics (2012 - No. 2)
Let x1, x2,........., xn be n observations, and let $$\overline x $$ be their arithematic mean and $${\sigma ^2}$$ be their variance.
Statement 1 : Variance of 2x1, 2x2,......., 2xn is 4$${\sigma ^2}$$.
Statement 2 : : Arithmetic mean of 2x1, 2x2,......, 2xn is 4$$\overline x $$.
Statement 1 : Variance of 2x1, 2x2,......., 2xn is 4$${\sigma ^2}$$.
Statement 2 : : Arithmetic mean of 2x1, 2x2,......, 2xn is 4$$\overline x $$.
Statement 1 is false, statement 2 is true
Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
Statement 1 is true, statement 2 is false
Explanation
Given that,
for $${x_1},{x_2},....{x_n},$$ $$A.M = \overline x $$
and variance $$ = {\sigma ^2}$$
Now A.M of
$$2{x_1},2x{}_2.....2{x_n} = {{2\left( {{x_1} + {x_2} + ....{x_n}} \right)} \over n} = 2\overline x $$
But given $$A.M = 4\overline x $$
$$\therefore\,\,\,$$ Statement $${\rm I}{\rm I}$$ is false.
Variance of $$2{x_1},2{x_2}......2{x_n}$$
$$=$$ Variance of $$\left\{ {2{x_i}} \right\}$$
$$ = {2^2}$$ Variance of $$\left\{ {{x_i}} \right\} = 4{\sigma ^2}$$
So, statement $${\rm I}$$ is correct.
for $${x_1},{x_2},....{x_n},$$ $$A.M = \overline x $$
and variance $$ = {\sigma ^2}$$
Now A.M of
$$2{x_1},2x{}_2.....2{x_n} = {{2\left( {{x_1} + {x_2} + ....{x_n}} \right)} \over n} = 2\overline x $$
But given $$A.M = 4\overline x $$
$$\therefore\,\,\,$$ Statement $${\rm I}{\rm I}$$ is false.
Variance of $$2{x_1},2{x_2}......2{x_n}$$
$$=$$ Variance of $$\left\{ {2{x_i}} \right\}$$
$$ = {2^2}$$ Variance of $$\left\{ {{x_i}} \right\} = 4{\sigma ^2}$$
So, statement $${\rm I}$$ is correct.
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