Question

The mean and variance of seven observation are 8 and 16 respectively .If five of these are 2,4,10,12and 14.Find the remaining 2 observations .

Answer 1

Let remaining two observations are x and y.
We know,
Mean = sum of observations/total number of observations
⇒8 = (2 + 4 + 10 + 12 + 14 + x + y)/7
⇒8 × 7 = 42 + (x + y)
⇒ 56 - 42 = (x + y)
⇒ (x + y) = 14 ------(1)

Variance of given observations is 14
we know, variance is given by
Variance = $\bold{\frac{\sum{(x_i-\overline{x})^2}}{n}}$
⇒16 = {(2 - 8)² + (4 - 8)² + (10 - 8)² + (12 - 8)² + (14 - 8)² + (x - 8)² + (y - 8)²}/7
⇒16 × 7 = (-6)² + (-4)² + (2)² + (4)² + (6)² + x² + y² - 16(x + y) + 8² + 8²
⇒112 = 36 + 16 + 4 + 16 + 36 + x² + y² - 16 × 14 + 64 + 64
⇒ 112 = 236 + x² + y² - 224
⇒112 = 12 + x² + y²
⇒x² + y² = 100 ------(2)

Solve equations (1) and (2),
x² + (14 - x)² = 100
⇒x² + 196 + x² - 28x = 100
⇒2x² -28x + 96 = 0
⇒x² - 14x + 48 = 0
⇒(x - 6)(x - 8) = 0⇒x = 6,8 put it in equation (1)
Then, y = 14 - 6 = 8, or y = 14 - 8 = 6

Hence, remaining two observations are 6, 8

Answer 2

Let  the  unknown observations x and  y

Observations:- 2, 4, 10 , 12, 14 , x , y

Total Number of  observation :  = 7

Sum of  observation = 2 + 4 + 10 + 12 + 14 + x + y = 42  + x + y

We know that ,

$(mean)\bar{x}=\frac{\text{Sum of obesevations}}{\text{Total Number of observations}}\\\\8=\frac{42+x+y}{7}\\\\x+y=56-42\\\\x+y=14.....................i)$

Calculate Variance:-

$\left|\begin{array}{ccc}x_i&d=x-\bar{x}&d^{2}\\2&-6&36\\4&-4&16\\10&2&4\\12&4&16\\14&6&36\\x&x-8&(x-8)^{2} \\y&y-8&(y-8)^{2}\end{array}\right|$

$\sum{d^{2}}=36+16+4+16+36+(x-8)^{2}+(y-8)^{2}\\\\=108+x^{2}+64-16x+y^{2}+64-16y\\ \\=236+x^{2}+y^{2}-16(x+y)\\\\=236+x^{2}+y^{2}-16\times14$

$Variance=\frac{\sum{d^{2}}}{n}\\\\16=\frac{x^{2}+y^{2}+12}{7}\\\\x^{2}+y^{2}+12=112\\\\x^{2}+y^{2}=100\\\\x^{2}+y^{2}+2xy-2xy=100\\\\(x+y)^{2}-2xy=100\\\\14^{2}-2xy=100\\ \\2xy=196-100\\\\2xy=96\\\\4xy=192\\\\(x-y)^{2}=(x+y)^{2} -4xy\\\\(x-y)^{2}=14^{2}-192\\ \\(x-y)^{2}=4\\\\x-y=4..................ii)$

Adding  equ i) and equ ii)

$2x=16\\\\x=8\\\\\text{Putting x=8 in equ i)}\\\\8+y=14\\\\y=6$

Hence,

That, observations are  8 and 6