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Q.

For a group containing 100 observation, the mean and
standard deviation are 8 and $\sqrt{10.5}$ For 50 observations
selected from these 100 observations, the mean
and standard deviation are 10 and 2 respectively . The
standard deviation of the other half is

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a

2

b

3

c

$\sqrt{6.5}$

d

(4)

answer is B.

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Detailed Solution

$n_{1} = n_{2} = 50, {\bar{x}}_{1} = 10, \bar{x} = 8, n = 100, σ = \sqrt{10.5}$

We find ${\bar{x}}_{2}$ and then $σ_{2}$

Using the combined mean formula, we have

$8 = \frac{50 \times 10 + 50 \times {\bar{x}}_{2}}{100}$

$\begin{array}{l} \Rightarrow {\bar{x}}_{2} = 6 \\ d_{1} = 10 - 8 = 2 and d_{2} = 6 - 8 = - 2 \\ 10.5 = σ^{2} = \frac{n_{1} (σ_{1}^{2} + d_{1}^{2}) + n_{2} (σ_{2}^{2} + d_{2}^{2})}{n_{1} + n_{2}} \\ = \frac{50 (4 + 4) + 50 (σ_{2}^{2} + 4)}{100} \\ σ_{2}^{2} = 9, σ_{2} = 3 \end{array}$

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