Let $\bar{x}$, $M$, and $\sigma^2$ denote the mean, mode, and variance, respectively, of the observations $x_1, x_2, \ldots, x_n$. Define $d_i = x_i - a$ for $i = 1, 2, \ldots, n$, where $a$ is any constant. Statement I: The variance of $d_1, d_2, \ldots, d_n$ is \(\sigma^2

Question

Let $\bar{x}$, $M$, and $\sigma^2$ denote the mean, mode, and variance, respectively, of the observations $x_1, x_2, \ldots, x_n$. Define $d_i = x_i - a$ for $i = 1, 2, \ldots, n$, where $a$ is any constant. Statement I: The variance of $d_1, d_2, \ldots, d_n$ is $\sigma^2$. Statement II: The mean and mode of $d_1, d_2, \ldots, d_n$ are $\bar{x} - a$ and $M - a$, respectively.

Accepted Answer

Both Statement I and Statement II are correct.. The variance of a set of observations is unaffected by adding or subtracting a constant, so the variance of the new set remains C3. Additionally, both the mean and mode shift by the same constant, resulting in C3 and M being adjusted by -a.

Solution

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