QJEEM12Maf784: JEE Main 2012 Mathematics - Class 11 JEEM12Maf784 Solution: Let x 1 , x 2 ,........., x n be n observations, and let be…

Question

Verified Solution • 2025–26 Session moderate

Let x₁, x₂,........., x_n be n observations, and let

\overline x

be their arithematic mean and

{\sigma ^2}

be their variance.

Statement 1 : Variance of 2x₁, 2x₂,......., 2x_n is 4

{\sigma ^2}

.
Statement 2 : : Arithmetic mean of 2x₁, 2x₂,......, 2x_n is 4

\overline x

.

A Statement 1 is false, statement 2 is true
B Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1
C Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1
D Statement 1 is true, statement 2 is false

Accepted Answer

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.

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