JEE MAIN - Mathematics (2023 - 8th April Evening Shift - No. 6)
Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$\mathrm{m}$$ and $$\mathrm{n}$$ are coprime, then $$\mathrm{m}+\mathrm{n}$$ is equal to :
317
316
314
315
Explanation
$$
\begin{aligned}
& \text { Since, Mean }=\frac{9}{2} \\\\
& \Rightarrow \Sigma x=\frac{9}{2} \times 12=54
\end{aligned}
$$
Also, variance $=4$
$$ \begin{aligned} & \Rightarrow \frac{\sum x^2}{12}=\left[\frac{\sum x_i}{12}\right]^2=4 \\\\ & \Rightarrow \frac{\sum x^2}{12}=4+\frac{81}{4}=\frac{97}{4} \\\\ & \Rightarrow \sum x^2=291 \end{aligned} $$
$$ \begin{aligned} & \sum x^{\prime}=54-(9+10)+7+14 \\\\ & =54-19+21=56 \\\\ & \text { and } \sum x^2=291-(81+100)+49+196 \\\\ & =291-181+49+196=355 \end{aligned} $$
$$ \begin{aligned} & \text { So, } \sigma_{\text {new }}^2=\frac{\sum x_{\text {new }}^2}{12}-\left(\frac{\sum x_{\text {new }}}{12}\right)^2 \\\\ & =\frac{355}{12}-\left(\frac{56}{12}\right)^2 \\\\ & =\frac{4260-3136}{144}=\frac{1124}{144}=\frac{281}{36} \\\\ & =\frac{m}{n} \\\\ & \Rightarrow m=281, n=36 \\\\ & \Rightarrow m+n=281+36=317 \end{aligned} $$
Also, variance $=4$
$$ \begin{aligned} & \Rightarrow \frac{\sum x^2}{12}=\left[\frac{\sum x_i}{12}\right]^2=4 \\\\ & \Rightarrow \frac{\sum x^2}{12}=4+\frac{81}{4}=\frac{97}{4} \\\\ & \Rightarrow \sum x^2=291 \end{aligned} $$
$$ \begin{aligned} & \sum x^{\prime}=54-(9+10)+7+14 \\\\ & =54-19+21=56 \\\\ & \text { and } \sum x^2=291-(81+100)+49+196 \\\\ & =291-181+49+196=355 \end{aligned} $$
$$ \begin{aligned} & \text { So, } \sigma_{\text {new }}^2=\frac{\sum x_{\text {new }}^2}{12}-\left(\frac{\sum x_{\text {new }}}{12}\right)^2 \\\\ & =\frac{355}{12}-\left(\frac{56}{12}\right)^2 \\\\ & =\frac{4260-3136}{144}=\frac{1124}{144}=\frac{281}{36} \\\\ & =\frac{m}{n} \\\\ & \Rightarrow m=281, n=36 \\\\ & \Rightarrow m+n=281+36=317 \end{aligned} $$
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