Consider the statistics of two sets of o | vedclass

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Question

$\sigma^{2}=\frac{ n _{1} \sigma_{1}^{2}+ n _{2} \sigma_{2}^{2}}{ n _{1}+ n _{2}}+\frac{ n _{1} n _{2}}{\left( n _{1}+ n _{2}\right)}\left(\overline{

Vedclass · Accepted Answer

$5$

Vedclass · Answer

$\sigma^{2}=\frac{ n _{1} \sigma_{1}^{2}+ n _{2} \sigma_{2}^{2}}{ n _{1}+ n _{2}}+\frac{ n _{1} n _{2}}{\left( n _{1}+ n _{2}ight)}\left(\overline{ x }_{1}-\overline{ x }_{2}ight)^{2}$

$n _{1}=10, n _{2}= n , \sigma_{1}^{2}=2, \sigma_{2}^{2}=1$

$\overline{ x }_{1}=2, \overline{ x }_{2}=3, \sigma^{2}=\frac{17}{9}$

$\frac{17}{9}=\frac{10 	imes 2+ n }{ n +10}+\frac{10 n }{( n +10)^{2}}(3-2)^{2}$

$\frac{17}{9}=\frac{(n+20)(n+10)+10 n}{(n+10)^{2}}$

$17 n^{2}+1700+340 n=90 n+9\left(n^{2}+30 n+200ight)$

$8 n^{2}-20 n-100=0$

$2 n^{2}-5 n-25=0$

$(2 n+5)(n-5)=0 \Rightarrow n=\frac{-5}{2} \,(Rejected) , 5$

Hence $n =5$

	Size	Mean	Variance
Observation $I$	$10$	$2$	$2$
Observation $II$	$n$	$3$	$1$

$\mathrm{x}$	$-2$	$-1$	$3$	$4$	$6$
$\mathrm{P}(\mathrm{X}=\mathrm{x})$	$\frac{1}{5}$	$\mathrm{a}$	$\frac{1}{3}$	$\frac{1}{5}$	$\mathrm{~b}$

Classes	$0-30$	$30-60$	$60-90$	$90-120$	$120-150$	$50-180$	$180-210$
$f_i$	$2$	$3$	$5$	$10$	$3$	$5$	$2$

Consider the statistics of two sets of observations as follows :

Size Mean Variance

Observation $I$ $10$ $2$ $2$

Observation $II$ $n$ $3$ $1$

If the variance of the combined set of these two observations is $\frac{17}{9},$ then the value of $n$ is equal to ..... .

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