The outcome of each of 30 items was observed , 10 items gave an outcome `(1)/(2)-d` each, 10 items gave outcome `(1)/(2)` each and the remaining 10 items gave outcome `(1)/(2) +d` each. If the variance of this outcome data is `(4)/(3)`, then |d| equals

Options

A. 2
B. `(sqrt(5))/(2)`
C. `(2)/(3)`
D. `sqrt(2)`

Answer

To solve the problem, we need to find the value of |d| given the outcomes of 30 items and the variance of these outcomes. Let's break it down step by step.

### Step 1: Define the outcomes
We have three groups of outcomes:
- 10 items gave an outcome of \( \frac{1}{2} - d \)
- 10 items gave an outcome of \( \frac{1}{2} \)
- 10 items gave an outcome of \( \frac{1}{2} + d \)

### Step 2: Calculate the mean (μ)
The mean (μ) of the outcomes can be calculated as follows:

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

Calculating this gives:

\[ \mu = \frac{1}{30} \left( 10 \left( \frac{1}{2} - d \right) + 10 \left( \frac{1}{2} \right) + 10 \left( \frac{1}{2} + d \right) \right) \]
\[ = \frac{1}{30} \left( 5 - 10d + 5 + 5 + 10d \right) \]
\[ = \frac{1}{30} \left( 15 \right) = \frac{1}{2} \]

### Step 3: Calculate the variance (σ²)
The variance is given by the formula:

\[ \sigma^2 = \frac{\sum (x_i^2)}{n} - \mu^2 \]

Where \( n = 30 \) and \( \sigma^2 = \frac{4}{3} \).

First, we calculate \( \sum (x_i^2) \):

\[ \sum (x_i^2) = 10 \left( \frac{1}{2} - d \right)^2 + 10 \left( \frac{1}{2} \right)^2 + 10 \left( \frac{1}{2} + d \right)^2 \]

Calculating each term:

1. \( 10 \left( \frac{1}{2} - d \right)^2 = 10 \left( \frac{1}{4} - d + d^2 \right) = \frac{10}{4} - 10d + 10d^2 \)
2. \( 10 \left( \frac{1}{2} \right)^2 = 10 \cdot \frac{1}{4} = \frac{10}{4} \)
3. \( 10 \left( \frac{1}{2} + d \right)^2 = 10 \left( \frac{1}{4} + d + d^2 \right) = \frac{10}{4} + 10d + 10d^2 \)

Combining these:

\[ \sum (x_i^2) = \frac{10}{4} - 10d + 10d^2 + \frac{10}{4} + \frac{10}{4} + 10d + 10d^2 \]
\[ = \frac{30}{4} + 20d^2 \]

### Step 4: Substitute into the variance formula
Now we substitute this back into the variance formula:

\[ \frac{\frac{30}{4} + 20d^2}{30} - \left( \frac{1}{2} \right)^2 = \frac{4}{3} \]

Calculating \( \left( \frac{1}{2} \right)^2 = \frac{1}{4} \):

\[ \frac{30 + 80d^2}{120} - \frac{1}{4} = \frac{4}{3} \]

Converting \( \frac{1}{4} \) to a fraction with a denominator of 120:

\[ \frac{30 + 80d^2}{120} - \frac{30}{120} = \frac{4}{3} \]

This simplifies to:

\[ \frac{80d^2}{120} = \frac{4}{3} + \frac{30}{120} \]

### Step 5: Solve for d
Now, we can solve for \( d^2 \):

\[ \frac{80d^2}{120} = \frac{4 + 30}{120} \]
\[ 80d^2 = 34 \quad \text{(after multiplying through by 120)} \]
\[ d^2 = \frac{34}{80} = \frac{17}{40} \]

### Step 6: Find |d|
Taking the square root of both sides:

\[ |d| = \sqrt{\frac{17}{40}} = \frac{\sqrt{17}}{\sqrt{40}} = \frac{\sqrt{17}}{2\sqrt{10}} = \frac{\sqrt{17}}{2\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{170}}{20} \]

Thus, the final answer is:

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