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The mean of the numbers $a, b, 8,5,10$ is 6 and the variance is $6.80$. Then which one of the following gives possible values of $a$ and $b$ ?
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Verified Answer
The correct answer is:
$a=3, b=4$
$a=3, b=4$
Mean of $a, b, 8,5,10$ is 6
$$
\begin{aligned}
& \Rightarrow \frac{a+b+8+5+10}{5}=6 \\
& \Rightarrow a+b=7
\end{aligned}
$$
Given that Variance is $6.8$
$$
\begin{aligned}
& \therefore \text { Variance }=\frac{\sum\left(\mathrm{X}_{\mathrm{i}}-\mathrm{A}\right)^2}{\mathrm{n}} \\
& =\frac{(\mathrm{a}-6)^2+(\mathrm{b}-6)^2+4+1+16}{5}=6.8
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow a^2+b^2=25 \\
& a^2+(7-a)^2=25 \\
& \Rightarrow a^2-7 a+12=0 \\
& \therefore a=4,3 \text { and } b=3,4
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow \frac{a+b+8+5+10}{5}=6 \\
& \Rightarrow a+b=7
\end{aligned}
$$
Given that Variance is $6.8$
$$
\begin{aligned}
& \therefore \text { Variance }=\frac{\sum\left(\mathrm{X}_{\mathrm{i}}-\mathrm{A}\right)^2}{\mathrm{n}} \\
& =\frac{(\mathrm{a}-6)^2+(\mathrm{b}-6)^2+4+1+16}{5}=6.8
\end{aligned}
$$
$$
\begin{aligned}
& \Rightarrow a^2+b^2=25 \\
& a^2+(7-a)^2=25 \\
& \Rightarrow a^2-7 a+12=0 \\
& \therefore a=4,3 \text { and } b=3,4
\end{aligned}
$$
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