Question

The mean of five observations is $4$ and their variance is $5.2$. If three of these observations are $1,2$ and $6$, then the other two are

• A. $2$ and $9$
• B. $3$ and $8$
• C. $4$ and $7$
• D. $5$ and $6$

Answer 1

The correct option is C

$4$ and $7$

Explanation for the correct option:

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

Since, it is given that the mean of $5$ observations is $4$ and three observations are $1,2$ and $6$.

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

Therefore,

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{5}{\sum \left(x\right)}}}}{5}=4 \\ \Rightarrow \underset{i=1}{\overset{5}{\sum \left(x\right)}}=20 \\ \Rightarrow 1+2+6+x_1+x_2=20 \\ \Rightarrow x_1+x_2=11 \\ \Rightarrow x_1=11-x_2...1 \end{gathered}$$

Also, it is given that the variance of $5$ observations is $5.2$.

$$\begin{gathered} \frac{{\displaystyle \underset{i=1}{\overset{5}{\sum \left(x^2\right)}}}}{5}-{\left(4\right)}^2=5.2 \\ \Rightarrow \frac{\underset{i=1}{\overset{5}{\sum \left(x^2\right)}}}{5}=21.2 \\ \Rightarrow \underset{i=1}{\overset{5}{\sum \left(x^2\right)}}=106 \\ \Rightarrow {\left(1\right)}^2+{\left(2\right)}^2+{\left(6\right)}^2+{\left(x_1\right)}^2+{\left(x_2\right)}^2=106 \\ \Rightarrow {\left(x_1\right)}^2+{\left(x_2\right)}^2=65...2 \end{gathered}$$

Step 2: Find the remaining two observations.

From equation $1$ and equation $2$.

$$\begin{gathered} {\left(11-x_2\right)}^2+x_2^2=65 \\ \Rightarrow {\left(x_2\right)}^2+121-22x_2+x_2^2=65 \\ \Rightarrow 2x_2^2-22x_2+56=0 \\ \Rightarrow x_2^2-11x_2+28=0 \\ \Rightarrow x_2=\frac{11\pm \sqrt{{11}^2-4\left(28\right)}}{2} \\ \Rightarrow x_2=\frac{11\pm \sqrt{121-112}}{2} \\ \Rightarrow x_2=\frac{11\pm \sqrt{9}}{2} \\ \Rightarrow x_2=\frac{11\pm 3}{2} \\ \Rightarrow x_2=4,7 \end{gathered}$$

For $x_2=4$,

$$\begin{gathered} x_1=11-4 \\ \Rightarrow x_1=7 \end{gathered}$$

For $x_2=7$,

$$\begin{gathered} x_1=11-7 \\ \Rightarrow x_1=4 \end{gathered}$$

Therefore, the remaining two observations are $4$ and $7$.

Hence, option $C$ is the correct option.