Let x_1, x_2, , x_ n be n observations, and let x be their arithematic mean and σ^2 be their variance. Statement 1 : Variance of 2 x_1, 2 x_2, , 2 x_n is 4 σ^2 . Statement 2 : Arithmetic mean of 2 x_1, 2 x_2, . ., 2 x_

Correct answer: D. Statement 1 is true, statement 2 is false σ^2=∑ (x_i^2)/(n)- (∑ (x_i)/(n))^2 Variance of 2 x_1, 2 x_2, . ., 2 x_n=∑ ((2 x_i)^2)/(n)-(∑ (2 x_i)/(n))^2=4[∑ (x_i^2)/(n)-(∑ (x_i)/(n))^2]=4 σ^2 Statement 1 is true. A.M. of 2 x_1, 2 x_2, , 2 x_n=(2 x_1+2 x_2+ +2 x_n)/(n)=2((x_1+x_2+ +x_n)/(n))=2 x Statement 2 is false.

JEE Main 2012Offline

Exam: JEE Main
Year: 2012
Mode: Offline
Subject: Mathematics
Chapter: Statistics

JEE Main 2012 · Offline · Mathematics · Statistics

Let x_1, x_2, , x_ n be n observations, and let x be their...

Let

\mathrm{x}_1, \mathrm{x}_2, \ldots \ldots, \mathrm{x}_{\mathrm{n}}

\mathrm{n}

observations, and let

\overline{\mathrm{x}}

be their arithematic mean and

\sigma^2

be their variance. Statement

1

: Variance of

2 x_1, 2 x_2, \ldots \ldots, 2 x_n

4 \sigma^2

. Statement

2

: Arithmetic mean of

2 \mathrm{x}_1, 2 \mathrm{x}_2, \ldots . ., 2 \mathrm{x}_{\mathrm{n}}

4 \overline{\mathrm{x}}

Options

AStatement $1$ is false, statement $2$ is true
BStatement $1$ is true, statement $2$ is true; statement $2$ is a correct explanation for statement $1$
CStatement $1$ is true, statement $2$ is true; statement $2$ is not a correct explanation for statement $1$
DStatement $1$ is true, statement $2$ is false

Correct answer

D. Statement $1$ is true, statement $2$ is false

Step-by-step solution

\sigma^2=\sum \frac{x_i^2}{n}-

\left(\sum \frac{x_i}{n}\right)^2

Variance of

2 x_1, 2 x_2, \ldots . ., 2 x_n=\sum \frac{\left(2 x_i\right)^2}{n}-\left(\sum \frac{2 x_i}{n}\right)^2=4\left[\sum \frac{x_i^2}{n}-\left(\sum \frac{x_i}{n}\right)^2\right]=4 \sigma^2

Statement

1

is true. A.M. of

2 x_1, 2 x_2, \ldots \ldots, 2 x_n=\frac{2 x_1+2 x_2+\cdots+2 x_n}{n}=2\left(\frac{x_1+x_2+\cdots+x_n}{n}\right)=2 \bar{x}

Statement

2

is false.

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