$\begin{array}{1 1}(A)\;8.94\\(B)\;5\\(C)\;7\\(D)\;15\end{array} $

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- The formula used is Combined $SD= \sqrt {\large\frac{n_1 \sigma_1^2+n_2 \sigma^2}{n_1+n_2} +\frac{n_1n_2 (\bar {x_1} -\bar {x_2})^2}{(n_1+n_2)^2}}$

Step 1:

Given $n_1=60 \qquad n_2=80$

$\bar{x_1}=650 \qquad \bar{x_2}=660$

$\sigma_1=8 \qquad \sigma_2=7$

The formula used is Combined $SD= \sqrt {\large\frac{n_1 \sigma_1^2+n_2 \sigma^2}{n_1+n_2} +\frac{n_1n_2 (\bar {x_1} -\bar {x_2})^2}{(n_1+n_2)^2}}$

$\qquad= \sqrt { \large\frac{60 \times 8^2 + 80 \times 7^2}{60+80}+ \frac{ 60 \times 80 (650-660)^2}{(60+80)^2}}$

$\qquad= \sqrt {\large\frac{ 3840+3920}{140}+ \frac{4800 \times 100}{(140)^2}}$

$\qquad= \sqrt {\large\frac{7760 \times 140+4800 \times 100}{(140^2)}}$

$\qquad= \large\frac{1}{140}$$ \sqrt {1086400+480000}$

$\qquad= \large\frac{1251.559}{140}$$=8.939$

$\qquad= 8.94$

Hence A is the correct answer.

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