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(A) Given: Mean $\mu = \frac{1}{50} \sum x_i = 16$ and Standard Deviation $\sigma = 16$.From the formula for standard deviation: $\sigma^2 = \frac{1}{

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$400$

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(A) Given: Mean $\mu = \frac{1}{50} \sum x_i = 16$ and Standard Deviation $\sigma = 16$.From the formula for standard deviation: $\sigma^2 = \frac{1}{50} \sum x_i^2 - \mu^2$.Substituting the values: $16^2 = \frac{1}{50} \sum x_i^2 - 16^2$.$\frac{1}{50} \sum x_i^2 = 16^2 + 16^2 = 256 + 256 = 512$.We need to find the mean of $(x_i - 4)^2$,which is $\frac{1}{50} \sum (x_i - 4)^2$.Expanding the expression: $\frac{1}{50} \sum (x_i^2 - 8x_i + 16) = \frac{1}{50} \sum x_i^2 - 8 \left( \frac{1}{50} \sum x_i \right) + \frac{1}{50} \sum 16$.Substituting the known values: $512 - 8(16) + 16 = 512 - 128 + 16 = 400$.

If both the mean and the standard deviation of $50$ observations $x_1, x_2, \dots, x_{50}$ are equal to $16$,then the mean of $(x_1 - 4)^2, (x_2 - 4)^2, \dots, (x_{50} - 4)^2$ is

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