The mean and variance of a set of 15 numbers are 12 and 14 respectively

Question

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 3 $σ^{2}$ is equal to

Smart Achievers · Accepted Answer

(3)

$\sum x_{i} = 15 \times 12 and \frac{\sum x_{i}^{2}}{15} - 12^{2} = 14, \sum y_{i} = 15 \times 14 and \frac{\sum y_{i}^{2}}{15} - 14^{2} = σ$

$Now 13 = \frac{(14 + 144) \times 15 + (σ^{2} + 196) \times 15}{30} - 13^{2} \Rightarrow 3 σ^{2} = 30$

Solution

Q.
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 3 $σ^{2}$ is equal to

1 33

2 36

3 30

4 27

Ans.
(3)

$\sum x_{i} = 15 \times 12 and \frac{\sum x_{i}^{2}}{15} - 12^{2} = 14, \sum y_{i} = 15 \times 14 and \frac{\sum y_{i}^{2}}{15} - 14^{2} = σ$

$Now 13 = \frac{(14 + 144) \times 15 + (σ^{2} + 196) \times 15}{30} - 13^{2} \Rightarrow 3 σ^{2} = 30$

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