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(C) Let the variance of observations $x_1, x_2, \dots, x_n$ be $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$.Let the new observations be $y_

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$a^2\sigma^2$

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(C) Let the variance of observations $x_1, x_2, \dots, x_n$ be $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$.Let the new observations be $y_i = ax_i$ for $i = 1, 2, \dots, n$.The mean of the new observations is $\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i = \frac{1}{n} \sum_{i=1}^n ax_i = a \left( \frac{1}{n} \sum_{i=1}^n x_i ight) = a\bar{x}$.The variance of the new observations is $	ext{Var}(y) = \frac{1}{n} \sum_{i=1}^n (y_i - \bar{y})^2$.Substituting $y_i = ax_i$ and $\bar{y} = a\bar{x}$ into the formula:$	ext{Var}(y) = \frac{1}{n} \sum_{i=1}^n (ax_i - a\bar{x})^2 = \frac{1}{n} \sum_{i=1}^n a^2(x_i - \bar{x})^2 = a^2 \left( \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 ight) = a^2\sigma^2$.

If the variance of observations $x_1, x_2, \dots, x_n$ is $\sigma^2$,then the variance of $ax_1, ax_2, \dots, ax_n$,where $a \neq 0$,is

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