Multiple choice

In a series of $100$ observations, half of them is equal to $p$ and remaining half equal to $-p$ if the standard deviation of these observations is $10$ then $|p|=$__

  1. $10$
  2. $\dfrac {1}{10}$
  3. $\dfrac {1}{\sqrt {10}}$
  4. $\sqrt {10}$
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A Correct answer
Explanation

For a set of 100 observations where 50 are p and 50 are -p, the mean is 0. The variance is the average of the squared deviations from the mean: (50 * p^2 + 50 * (-p)^2) / 100 = (50p^2 + 50p^2) / 100 = 100p^2 / 100 = p^2. Since standard deviation is 10, variance is 100, so p^2 = 100 and |p| = 10.

AI explanation

The mean of the 100 observations is calculated as (50 multiplied by p plus 50 multiplied by negative p) divided by 100, which equals 0. The variance is the average of the squared deviations, becoming (50 times p squared plus 50 times p squared) divided by 100, simplifying to p squared. Since the standard deviation is given as 10, the variance is 100, making the absolute value of p equal to 10.

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