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​

Answer 1

Answer:

The Remaining 2 observations = 6 and 8      

Step-by-step explanation:

Given,

Mean (= 8

Variance = 16

No. of observations = 7

Let the remaining two observations = x and y

∴ Observations = 2, 4, 10, 12, 14, x, y

Since,

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

⇒ 8 = $\frac{42+x+y}{7}$

⇒ 56 = 42 + x + y

⇒ x + y = 56 - 42

⇒ x + y = 14                equation ( i )

and

  Variance = $\frac{1}{n}$ ∑($x_{i} - Mean$)

⇒ 16 = $\frac{1}{7}$ [( -6 )² + ( -4 )² + ( 2 )² + ( 4 )² + ( 6 )² + x² + y² - 2 × 8 ( x + y ) + 2 × ( 8 )² ]

⇒ 16 × 7 =  36 + 16 + 4 + 6 + 36 + x² + y² - 16 ( x + y ) + 2 ( 64 )    

⇒ 112 = 108 + x² + y² - 16 ( 14 ) + 128

⇒  x² + y² = 112 - 108 + 224 - 128

⇒  x² + y² = 100                                equation ( ii )

Squaring equation ( i ), we obtain

⇒ ( x + y )² = ( 14 )²

⇒  x² + y² + 2xy = 196

⇒  x² + y² = 196 - 2xy                      equation ( iii )

From equation ( ii ) and ( iii )

⇒ 196 - 2xy = 100

⇒ 2xy = 196 - 100

⇒ 2xy = 96                                   equation ( iv )

Subtract equation ( iv ) from equation ( ii )

⇒ x² + y² - 2xy = 100 - 96

⇒ ( x - y )² = 4

⇒ x - y = ± 2

Now,

When  x - y = 2

∴ x = 8    and   y = 6

When  x - y = - 2

∴ x = 6    and   y = 8

Therefore,

The Remaining 2 observations = 6 and 8