Question

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

• A. $2$
• B. $\frac{\sqrt{5}}{2}$
• C. $\frac{2}{3}$
• D. $\sqrt{2}$

Answer 1

The Correct Option is D

To solve this problem, we need to find the absolute value of \(d\) such that the variance of the outcomes is \(\frac{4}{3}\). 

The outcomes for the 30 items are given as:

• 10 items with outcome \(\frac{1}{2} - d\)
• 10 items with outcome \(\frac{1}{2}\)
• 10 items with outcome \(\frac{1}{2} + d\)

 

The mean of these outcomes, \(\overline{x}\), can be calculated as follows:

\[ \overline{x} = \frac{1}{30} \left(10 \left(\frac{1}{2} - d\right) + 10 \left(\frac{1}{2}\right) + 10 \left(\frac{1}{2} + d\right) \right) \]

\[ = \frac{1}{30} \left( 10 \times \frac{1}{2} - 10d + 10 \times \frac{1}{2} + 10 \times \frac{1}{2} + 10d \right) \]

\[ = \frac{1}{30} \left( 5 + 0 + 5 \right) = \frac{1}{2} \]

The variance is given by the formula:

\[ \text{Variance} = \frac{1}{30} \left(10 \left(\frac{1}{2} - d - \frac{1}{2}\right)^2 + 10 \left(\frac{1}{2} - \frac{1}{2}\right)^2 + 10 \left(\frac{1}{2} + d - \frac{1}{2}\right)^2 \right) \]

Simplifying each term:

\[ \left(\frac{1}{2} - d - \frac{1}{2}\right)^2 = d^2 \]

\[ \left(\frac{1}{2} - \frac{1}{2}\right)^2 = 0 \]

\[ \left(\frac{1}{2} + d - \frac{1}{2}\right)^2 = d^2 \]

Substituting these back into the variance formula:

\[ \text{Variance} = \frac{1}{30} \left(10d^2 + 0 + 10d^2 \right) = \frac{20d^2}{30} = \frac{2d^2}{3} \]

We are given that the variance is \(\frac{4}{3}\):

\[ \frac{2d^2}{3} = \frac{4}{3} \]

This simplifies to:

\[ 2d^2 = 4 \]

\[ d^2 = 2 \]

\[ |d| = \sqrt{2} \]

Therefore, the correct answer is \(\sqrt{2}\).