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$6$ અવલોકનો $a$, $b,$ $68,$ $44,$ $48,$ $60$ ના મધ્યક અને વિચરણ અનુક્કમે $55$ અને $194$ છે. જો $a > b,$ તો $a +$ $3 b=$..........................
$200$
$190$
$180$
$210$
Solution
$\mathrm{a}, \mathrm{b}, 68,44,48,60$
Mean $=55$ $a>b$
Variance $=194$ $a+3 b$
$\frac{a+b+68+44+48+60}{6}=55$
$\Rightarrow 220+a+b=330$
$\therefore a+b=110 \ldots . .(1)$
Also,
$\sum \frac{\left(\mathrm{x}_{\mathrm{i}}-\overline{\mathrm{x}}\right)^2}{\mathrm{n}}=194 $
$\Rightarrow(\mathrm{a}-55)^2+(\mathrm{b}-55)^2+(68-55)^2+(44-55)^2$
$+(48-55)^2+(60-55)^2=194 \times 6$
$\Rightarrow(\mathrm{a}-55)^2+(\mathrm{b}-55)^2+169+121+49+25=1164$
$\Rightarrow(\mathrm{a}-55)^2+(\mathrm{b}-55)^2=1164-364=800$
$\mathrm{a}^2+3025-110 \mathrm{a}+\mathrm{b}^2+3025-110 \mathrm{~b}=800$
$\Rightarrow \mathrm{a}^2+\mathrm{b}^2=800-6050+12100$
${a}^2+\mathrm{b}^2=6850 \ldots \ldots .(2)$
Solve $(1) \& (2);$
$a=75, b=35$
$\therefore$ $a+3 b=75+3(35)=75+105=180$
