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.

Marks	$10-20$	$20-30$	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$
Group $A$	$9$	$17$	$32$	$33$	$40$	$10$	$9$
Group $B$	$10$	$20$	$30$	$25$	$43$	$15$	$7$

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

The variance of the first $n$ natural numbers is

The standard deviation of the numbers $31, 32, 33, \ldots, 46, 47$ is

From the data given below,state which group is more variable,$A$ or $B$?

Marks $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ $70-80$

Group $A$ $9$ $17$ $32$ $33$ $40$ $10$ $9$

Group $B$ $10$ $20$ $30$ $25$ $43$ $15$ $7$

If each of the observations $x_{1}, x_{2}, \ldots, x_{n}$ is increased by $a$,where $a$ is a positive or negative number,show that the variance remains unchanged.

The coefficient of variation for the following data is
Class interval $0-2$ $2-4$ $4-6$ $6-8$ $8-10$
Frequency $2$ $3$ $5$ $3$ $2$

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 $4\bar x$.