Mathematics - Statistics Question with Solution | TestHub

Given,

Mean and variance of $7$ observation is given as $8 \& 16$ respectively,

So using the formula of mean we get, $\overline{x}=\frac{S}{7}\Rightarrow S=8\times 7=56$

Now if we remove $14$ sum will be $S=56-14=42$ and new mean will be $a=\frac{42}{6}=7$

Now using the formula of variance we get,

$\frac{{\displaystyle \sum _{i=1}^6{x}_i^2+{14}^2}}{7}-{\left(\overline{x}\right)}^2=16$

$\Rightarrow \frac{{\displaystyle \sum _{i=1}^6{x}_i^2+{14}^2}}{7}-8^2=16$

$\Rightarrow \frac{{\displaystyle \sum _{i=1}^6{x}_i^2+{14}^2}}{7}=80$

$\Rightarrow \sum _{i=1}^6{x}_i^2+{14}^2=560$

$\Rightarrow \sum _{i=1}^6{x}_i=364$

Now, new variance $b=\frac{\sum _{i=1}^6{x}_i^2}{6}-a^2=\frac{364}{6}-7^2=\frac{35}{3}$

Hence, the value of $a+3b-5=7+35-5=37$