From the given data,

Variance $=7$, the number of observations $\left(n\right)=50$.

Variance $=\frac{1}{n}\sum {\left(x_i-\overline{x}\right)}^2$

$\Rightarrow 7=\frac{1}{50}\sum {\left(x_i-\overline{x}\right)}^2$

$\Rightarrow 350=\sum {\left(x_i-\overline{x}\right)}^2 ...\left(1\right)$ 

Each observation is multiplied by $6$ and then $5$ is subtracted.

Let the new observations be $y_1, y_2, y_3,\dots ,y_{50}$.

$y_i=6x_i-5$

$x_i=\frac{y_i+5}{6} ...\left(2\right)$ 

The new variance $=\frac{1}{n}\sum {\left(y_i-\overline{y}\right)}^2$

$\overline{y}=\frac{1}{n}\sum y_i ...\left(3\right)$ 

$\Rightarrow \overline{y}=\frac{1}{50}\sum \left(6x_i-5\right)$

$\Rightarrow \overline{y}=\frac{1}{50}\sum 6x_i-\sum _0^{50}\frac{1}{50}·5$

$\Rightarrow \overline{y}=\frac{1}{50}\sum 6x_i-5$

$\Rightarrow \overline{y}=6\overline{x}-5$   [$\therefore$ From $\left(1\right)$]

$\Rightarrow \overline{x}=\frac{\overline{y}+5}{6} ...\left(4\right)$ 

$\Rightarrow \sum {\left(x_i-\overline{x}\right)}^2=350$

$\Rightarrow \sum {\left(\frac{y_i+5}{6}-\frac{\overline{y}+5}{6}\right)}^2=350$  [$\therefore$ From $\left(2\right) \& \left(4\right)$]

$\Rightarrow \sum {\left(\frac{1}{6}\right)}^2{\left(y_i-\overline{y}\right)}^2=350$

$\Rightarrow \sum {\left(y_i-\overline{y}\right)}^2=36\times 350$

$\Rightarrow \sum {\left(y_i-\overline{y}\right)}^2=12600$

New variance$=\frac{1}{50}\sum {\left(y_i-\overline{y}\right)}^2=\frac{1}{50}\times 12600=252$