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JEE 2023 Mathematics PYQ — The mean and variance of a set of 15 numbers are 12 and 14 respec… | Mathem Solvex | Mathem Solvex

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and $σ^{2}$ respectively. If the variance of all the 30 numbers in the two sets is 13, then $σ^{2}$ is equal to:

Choose the correct answer:

A.
12
B.
10
(Correct Answer)
C.
11
D.
9

Correct Answer:

Explanation

Combined variance = $\frac{n _{1} σ ^{2} + n _{2} σ ^{2}}{n _{1} + n _{2}} + \frac{n _{1} n _{2} ( m _{1} - m _{2} ) ^{2}}{( n _{1} + n _{2} ) ^{2}}$

$13 = \frac{15.14 + 15. σ ^{2}}{30} + \frac{15 \times 15 ( 12 - 14 ) ^{2}}{30 \times 30}$

$σ^{2} = 10$

Explanation

Combined variance = $\frac{n _{1} σ ^{2} + n _{2} σ ^{2}}{n _{1} + n _{2}} + \frac{n _{1} n _{2} ( m _{1} - m _{2} ) ^{2}}{( n _{1} + n _{2} ) ^{2}}$

$13 = \frac{15.14 + 15. σ ^{2}}{30} + \frac{15 \times 15 ( 12 - 14 ) ^{2}}{30 \times 30}$

$σ^{2} = 10$

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