Let {x_1}, {x_2}, ldots, {x_n} be n observati | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given observations are ${x_1}, {x_2}, \ldots, {x_n}$ with mean $\bar x$ and variance ${\sigma ^2} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2$.F

Vedclass · Accepted Answer

Statement-$1$ is true,Statement-$2$ is false.

Vedclass · Answer

(D) Given observations are ${x_1}, {x_2}, \ldots, {x_n}$ with mean $\bar x$ and variance ${\sigma ^2} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2$.For Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $\frac{1}{n} \sum_{i=1}^{n} (2x_i) = 2 \left( \frac{1}{n} \sum_{i=1}^{n} x_i \right) = 2\bar x$.Since the statement claims the mean is $4\bar x$,Statement-$2$ is false.For Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $\frac{1}{n} \sum_{i=1}^{n} (2x_i - 2\bar x)^2 = \frac{1}{n} \sum_{i=1}^{n} 4(x_i - \bar x)^2 = 4 \left( \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar x)^2 \right) = 4{\sigma ^2}$.Thus,Statement-$1$ is true.Therefore,Statement-$1$ is true and Statement-$2$ is false.

Class Interval	Frequency
$0 - 6$	$10$
$6 - 12$	$8$
$12 - 18$	$6$
$18 - 24$	$4$
$24 - 30$	$2$

Marks of $A$	$30, 20, 40$
Marks of $B$	$70, 0, 5$

Class Interval	$0-10$	$10-20$	$20-30$	$30-40$
Frequency	$2$	$3$	$4$	$1$

Let ${x_1}, {x_2}, \ldots, {x_n}$ be $n$ observations,and let $\bar x$ be their arithmetic mean and ${\sigma ^2}$ be the variance.
Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4{\sigma ^2}$.
Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4\bar x$.

Similar Questions

If each observation of a distribution with variance $\sigma^2$ is multiplied by $\lambda$,find the standard deviation of the new observations.

The variance of the following frequency distribution is
Class Interval Frequency
$0 - 6$ $10$
$6 - 12$ $8$
$12 - 18$ $6$
$18 - 24$ $4$
$24 - 30$ $2$

The marks obtained by students $A$ and $B$ in $3$ examinations are given below:

Marks of $A$ $30, 20, 40$

Marks of $B$ $70, 0, 5$

The ratio of the coefficient of variation of marks of $A$ and the coefficient of variation of marks of $B$ is:

The variance of the following continuous frequency distribution is
Class Interval $0-10$ $10-20$ $20-30$ $30-40$
Frequency $2$ $3$ $4$ $1$

If the mean and variance of the following data: $6, 10, 7, 13, a, 12, b, 12$ are $9$ and $\frac{37}{4}$ respectively,then $(a-b)^{2}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let ${x_1}, {x_2}, \ldots, {x_n}$ be $n$ observations,and let $\bar x$ be their arithmetic mean and ${\sigma ^2}$ be the variance.Statement-$1$: The variance of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4{\sigma ^2}$.Statement-$2$: The arithmetic mean of $2{x_1}, 2{x_2}, \ldots, 2{x_n}$ is $4\bar x$.