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

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(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

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

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(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.

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 $4\bar x$.

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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 $4\bar x$.