Let bar{x}, M and sigma^2 be respectively the | vedclass

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(B) Given $d_i = -x_i - a$. Statement $I$: The variance of a set of observations is invariant under change of origin and scale factor $-1$. Specifical

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Statement $I$ and Statement $II$ are both true

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(B) Given $d_i = -x_i - a$. Statement $I$: The variance of a set of observations is invariant under change of origin and scale factor $-1$. Specifically,if $y_i = c x_i + k$,then $	ext{Var}(y) = c^2 	ext{Var}(x)$. Here,$c = -1$ and $k = -a$. Thus,$	ext{Var}(d) = (-1)^2 \sigma^2 = \sigma^2$. So,Statement $I$ is true. Statement $II$: The mean of $d_i$ is $\bar{d} = \frac{1}{n} \sum (-x_i - a) = -\bar{x} - a$. This is correct. For the mode,if $M$ is the mode of $x_i$,then the mode of $d_i = -x_i - a$ is $-M - a$. This is also correct. Therefore,both statements are true.

$x_i$	$1$	$2$	$3$	$4$	$5$
$f_i$	$5$	$4$	$f$	$2$	$3$

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