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 \(\sigma^{2}\) respectively. If the variance of all the 30 numbers in the two sets is 13 , then \(\sigma^{2}\) is equal to

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Statistics

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

Answer: (D)

The combined variance formula is

σ_{co mbin e d}^{2} = \frac{n _{1} σ _{1}^{2} + n _{2} σ _{2}^{2}}{n _{1} + n _{2}} + \frac{n _{1} n _{2} ( m _{1} - m _{2} ) ^{2}}{( n _{1} + n _{2} ) ^{2}}

. Here

n_{1} = n_{2} = 15

m_{1} = 12

m_{2} = 14

σ_{1}^{2} = 14

σ_{2}^{2} = σ^{2}

, and

σ_{co mbin e d}^{2} = 13

. Substituting:

13 = \frac{15 \cdot 14 + 15 σ ^{2}}{30} + \frac{15 \cdot 15 ( 12 - 14 ) ^{2}}{900} = \frac{14 + σ ^{2}}{2} + \frac{15 \cdot 15 \cdot 4}{900} = \frac{14 + σ ^{2}}{2} + 1

. Solving:

13 = \frac{14 + σ ^{2}}{2} + 1

, so

12 = \frac{14 + σ ^{2}}{2}

, giving

24 = 14 + σ^{2}

and

σ^{2} = 10

JEE Main · 2023 · 06 Apr 2023 · Shift 1 · Single correct option