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

Correct answer: D. 2 Outcomes are ((1)/(2)-d),((1)/(2)-d), 0 , 10 times, (1)/(2), (1)/(2) , , 10 times, (1)/(2)+d, (1)/(2)+d, , 10 times Mean =(1)/(30)((1)/(2) × 30)= (1)/(2) Variance of the outcomes is, aligned σ^ 2 &=(1)/(30) _i^ 2 -( x )^ 2 &=(1)/(30)[((1)/(2)-d)^ 2 ×₁₀⁺((1)/(2))^ 2 ×₁₀⁺((1)/(2)+d)^ 2 ×10]-(1)/(4) & (4)/(3)=(1)/(30)[30 × (1)/(4)+20 d^ 2 ]-(1)/(4) & (4)/(3)=(1)/(4)+(2)/(3)d^ 2 -(1)/(4) & d^ 2 =2 |d|= 2 aligned

JEE Main 201911 JanuaryMorning Shift

Exam: JEE Main
Year: 2019
Date: 11 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Statistics

JEE Main 2019 · 11 January · Morning Shift · Mathematics · Statistics

The outcome of each of 30 items was observed; 10 items gave an...

The outcome of each of 30 items was observed; 10 items gave an outcome

\frac{1}{2}-\mathrm{d}

each, 10 items gave outcome

\frac{1}{2}

each and the remaining 10 items gave outcome

\frac{1}{2}+\mathrm{d}

each. If the variance of this outcome data is

\frac{4}{3}

then

|\mathrm{d}|

equals:

Options

A $\frac{2}{3}$
B2
C $\frac{\sqrt{5}}{2}$
D $\sqrt{2}$

Correct answer

D. $\sqrt{2}$

Step-by-step solution

Outcomes are

\left(\frac{1}{2}-d\right),\left(\frac{1}{2}-d\right), 0 \ldots, 10

times,

\frac{1}{2}, \frac{1}{2}

\ldots, 10

times,

\frac{1}{2}+d, \frac{1}{2}+d, \ldots, 10

times

\text{ Mean }=\frac{1}{30}\left(\frac{1}{2} \times

30\right)=

\frac{1}{2}

Variance of the outcomes is,

\begin{aligned}\sigma^{2}&=\frac{1}{30} \Sigmax_i^{2}-(\bar{x})^{2} \\ &=\frac{1}{30}\left[\left(\frac{1}{2}-d\right)^{2} \times₁₀⁺\left(\frac{1}{2}\right)^{2} \times₁₀⁺\left(\frac{1}{2}+d\right)^{2} \times10\right]-\frac{1}{4}\\ \Rightarrow & \frac{4}{3}=\frac{1}{30}\left[30 \times \frac{1}{4}+20 d^{2}\right]-\frac{1}{4}\\ \Rightarrow & \frac{4}{3}=\frac{1}{4}+\frac{2}{3}d^{2}-\frac{1}{4}\\ \Rightarrow & d^{2}=2 \Rightarrow|d|=\sqrt{2} \end{aligned}

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