Multiple choice

In a set of $2n$ observations, half of them are equal to '$\alpha$' and the remaining half are equal to '$-\alpha$'. If the standard deviation of all the observations is $2$, then the value of $|\alpha |$ is equal to

  1. $2$
  2. $\sqrt 2$
  3. $2 \sqrt 2$
  4. $4$
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A Correct answer
Explanation

For a set of 2n observations with n equal to alpha and n equal to -alpha, the mean is 0. Variance = [n(alpha^2) + n(-alpha^2)] / 2n = (2n * alpha^2) / 2n = alpha^2. Standard deviation = sqrt(variance) = |alpha|. Given SD = 2, |alpha| = 2.

AI explanation

The mean of the 2n observations is 0 because the values alpha and minus alpha cancel each other out. Using the standard deviation formula, the variance is the average of the squared deviations, giving the equation 4 equals the sum of n times alpha squared plus n times alpha squared, divided by 2n. This simplifies to 4 equals alpha squared, meaning the absolute value of alpha is 2.

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