Multiple choice

The outcome of each of $30$ items was observed; $10$ items gave an outcome $\dfrac{1}{2}$- d each, $10$ items gave outcome $\dfrac{1}{2}$ each and the remaining $10$ items gave outcome $\dfrac{1}{2}+$ d each. If the variance of this outcome data is $\dfrac{4}{3}$ then $|d|$ equals:-

  1. $2$
  2. $\dfrac{\sqrt5}{2}$
  3. $\dfrac{2}{3}$
  4. $\sqrt2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The three values are 1/2 - d, 1/2, and 1/2 + d, each occurring 10 times. Their mean is 1/2, so the variance is (d^2 + 0 + d^2)/3 = 2d^2/3. Equating this to 4/3 gives d^2 = 2, so |d| = sqrt(2).

AI explanation

The mean of the 30 items is calculated as (10 times (0.5 minus d) plus 10 times 0.5 plus 10 times (0.5 plus d)) divided by 30, which equals 0.5. The variance is found by the average of the squared deviations from the mean, giving 10 times ((minus d) squared plus 0 squared plus d squared) divided by 30. Setting the variance equal to 4 divided by 3 results in 20 times d squared divided by 30 equals 4 divided by 3, meaning d squared equals 2. Taking the absolute value gives |d| equals square root of 2.

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