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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 <img alt="\sqrt{13.44}" src="https://entrancecorner.codecogs.com/gif.latex?%5Csqrt%7B13.44

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 \sqrt{13.44}, then the standard deviation of the second sample is:
Option: 1 8
Option: 2 6
Option: 3 4
Option: 4 5

Answers (1)

best_answer

n_{1}= 100, n_{1}+n_{2}= 250\Rightarrow n_{2}= 150
\bar{x_{1}}= 15,\bar{x}= 15\cdot 6
Now\, \bar{x}= \frac{n_{1}\bar{x_{1}}+n_{2}\bar{x_{2}}}{n_{1}+n_{2}}
\Rightarrow 15\cdot 6= \frac{100\left ( 15 \right )+150\bar{x_{2}}}{250}
\Rightarrow \bar{x_{2}}= 16

SD_{1}= 3\Rightarrow \sigma ^{2}1= 9

SD= \sqrt{13\cdot 44}\Rightarrow \sigma ^{2}= 13\cdot 44
Now\, \sigma ^{2}=\frac{n_{1}\left ( \sigma ^{2}_{1} +\left ( \bar{x}_{1}-\bar{x} \right )^{2}\right )+n_{2}\left ( \sigma ^{2}_{2}+\left (\bar{x}_{2} -\bar{x} \right )^{2} \right )}{n_{1}+n_{2}}
\Rightarrow 13\cdot 44= \frac{100\left ( 9+0\cdot 6^{2} \right )+150\left ( \sigma ^{2}_{2}+0\cdot 4^{2} \right )}{250}
\Rightarrow \sigma ^{2}_{2}= 16\Rightarrow \sigma _{2}= 4

Posted by

Kuldeep Maurya

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