The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean 15 . 6 and standard deviation 13 . 44 , then the standard devia

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The first of the two samples in a group has 100 items with mean 15 and standard deviation 3 . If the whole group has 250 items with mean 15 . 6 and standard deviation 13 . 44 , then the standard deviation of the second sample is:

TestHub · Accepted Answer

We have, number of terms in the first group $n_{1} = 100$ with their mean $X_{1} = 15$ and number of terms in whole group is $n_{1} + n_{2} = 250$ and their mean $X = 15.6 .$

Variance of first group $V_{1} (x) = 9$ and whole group is $Var (x) = 13.44 .$

Now, $σ^{2} = \frac{n_{1} σ_{1}^{2} + n_{2} σ_{2}^{2}}{n_{1} + n_{2}} + \frac{n_{1} n_{2}}{{(n_{1} + n_{2})}^{2}} {({\bar{x}}_{1} - {\bar{x}}_{2})}^{2}$

and $n_{2} = 150, {\bar{x}}_{2} = 16, V_{2} (x) = σ_{2}$

Therefore, $13.44 = \frac{100 \times 9 + 150 \times σ_{2}^{2}}{250} + \frac{100 \times 150}{{(250)}^{2}} \times 1$

$\Rightarrow σ_{2}^{2} = 16$

$\Rightarrow σ_{2} = 4$

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