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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 remaining 10 items gave outcome $\frac{1}{2}+d$ each. If the variance of this outcome data is $\frac{4}{3}$ then $\left |d \right |$ equals:
Option 1)
2
Option 2)

$\frac{2}{3}$
Option 3)
$\frac{\sqrt5}{2}$

Option 4)
$\sqrt2$

Answers (1)

best_answer

Variance -

In case of discrete data

$\dpi{100} \sigma ^{2}= \left ( \frac{\sum x_{i}^{2}}{n} \right )-\left ( \frac{\sum x_{i}}{n} \right )^{2}$

-

$\sigma ^{2}=\frac{\Sigma x^{2}}{N}-\mu ^{2}=\frac{4}{3}$

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

$=\frac{1}{2}$

$\sigma ^{2}=\frac{10\times \left ( \frac{1}{2}+d \right )^{2}+10\times \frac{1}{4}+10(\frac{1}{2}-d)^{2}}{30}-\frac{1}{4}$

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

$\Rightarrow \sigma ^{2}=\frac{1}{3}\left [ 2\left ( \frac{1}{4}+d^{2} +\frac{1}{4}\right ) \right ]-\frac{1}{4}$

$=\frac{1}{3}\left [ \frac{3}{4}+2d^{2}\right ]-\frac{1}{4}$

$=\frac{2d^{2}}{3}$

Now, $\sigma ^{2}=\frac{4}{3}$

$\Rightarrow \frac{2d^{2}}{3}=\frac{4}{3}\Rightarrow |d|=\sqrt{2}$

Option 1)

2

Option 2)

$\frac{2}{3}$

Option 3)

$\frac{\sqrt5}{2}$

Option 4)
$\sqrt2$

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