Question

The mean and variance of $8$ observations are $10$ and $13.5$, respectively. If $6$ of these observations are $5,7,10,12,14,15$ then the absolute difference of the remaining two observations is:

• A. $3$
• B. $9$
• C. $7$
• D. $5$

Answer 1

The correct option is C

$7$

Explanation for the correct option:

Step 1: Find the relation between the remaining two observations.

Since, it is given that the mean of $8$ observations is $10$ and six observations are $5,7,10,12,14,15$.

Assume that, the remaining two observations are $x_1$ and $x_2$.

Therefore,

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{8}{\sum \left(x\right)}}}}{8}=10 \\ \Rightarrow \underset{i=1}{\overset{8}{\sum \left(x\right)}}=80 \\ \Rightarrow 5+7+10+12+14+15+x_1+x_2=80 \\ \Rightarrow x_1+x_2=17...1 \end{gathered}$$

Also, it is given that the variance of $8$ observations is $13.5$.

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{8}{\sum \left(x^2\right)}}}}{8}-{\left(10\right)}^2=13.5 \\ \Rightarrow \frac{\underset{i=1}{\overset{8}{\sum \left(x^2\right)}}}{8}=113.5 \\ \Rightarrow \underset{i=1}{\overset{8}{\sum \left(x^2\right)}}=908 \\ \Rightarrow {\left(5\right)}^2+{\left(7\right)}^2+{\left(10\right)}^2+{\left(12\right)}^2+{\left(14\right)}^2+{\left(15\right)}^2+{\left(x_1\right)}^2+{\left(x_2\right)}^2=908 \\ \Rightarrow {\left(x_1\right)}^2+{\left(x_2\right)}^2=169...2 \end{gathered}$$

Step 2: Find the absolute difference between the remaining two observations.

Since, we know that ${(x_1+x_2)}^2-2x_1{x}_2=x_1^2+x_2^2$

From equation $1$ and equation $2$.

$$\begin{gathered} {\left(17\right)}^2-2x_1{x}_2=169 \\ \Rightarrow 289-2x_1{x}_2=169 \\ \Rightarrow 2x_1{x}_2=120...3 \end{gathered}$$

Since, ${\left(x_1-x_2\right)}^2={(x_1+x_2)}^2-4x_1{x}_2$.

From equation $1$ and equation $3$.

$$\begin{gathered} {\left(x_1-x_2\right)}^2={\left(17\right)}^2-2·120 \\ \Rightarrow {\left(x_1-x_2\right)}^2=289-240 \\ \Rightarrow {\left(x_1-x_2\right)}^2=49 \\ \Rightarrow \left|x_1-x_2\right|=7 \end{gathered}$$

Therefore, the absolute difference between the remaining two observations is $7$.

Hence, option $C$ is the correct answer.