Question

The mean and variance of $7$ observations are $8$ and $16$ respectively. If $5$ of the observations are $2,4,10,12,14,$ find the remaining two observations.

• A. $3,6$
• B. $6,8$
• C. $1,5$
• D. None of these

Answer 1

The correct option is B $6,8$

Let $x$ and $y$ be the remaining two observations.

Then,

Mean $=8$

$\frac{2+4+10+12+14+x+y}{7}=8$

$42+x+y=56$

$x+y=14$

Variance $=16$

$\frac{1}{7}(2^2+4^2+{10}^2+{12}^2+{14}^2+x^2+y^2)-(Mean{)}^2=16$

$460+x^2+y^2=7\times 80$

$x^2+y^2=100$

Now, $(x+y{)}^2+(x-y{)}^2=2(x^2+y^2)$

$(x-y{)}^2=4$

$x-y=\pm 2$

If $x-y=2,x+y=14$, we get, $x=8,y=6$.

If $x-y=-2,x+y=14$, we get, $x=6,y=8$

Hence, the remaining two observations are 6 and 8.