Question

The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.

Answer 1

Let those two numbers be x and y.

∴ `overline x = 9 = (6 + 7 + 10 + 12 + 12 + 13 + x + y)/8`

or 72 = 60 + x + y

∴ x + y = 12      ...(i)

Variance σ² = `1/n^2 [nsumx_i^2 - (sumx_i)^2]`

∴ `sumx _i = 9`

∴ `9.25 = 1/64 [8 xx sumx_i^2 - (sumx_i)^2]`

∴ `sumx_i = 72`

`8 xx sumx_i = 9.25 xx 64 + 72 xx 72`

= 592 + 5184

= 5776

∴ `sumx_i^2 = 5776/8`

= 722

or `sumx_i^2 = 722`

= 6² + 7² + 10² + 12² + 12² + 13² + x² + y²

722 = 36 + 49 + 100 + 144 + 144 + 169 + x² + y²

= 642 + x² + y²

x² + y²

= 722 – 642

= 80

∴ x² + y² = 80        ....(ii)

From equations (i) and (ii)

or x² + (12 – x)² = 80

or 2x² – 24x + 144 = 80

or x² – 12x + 32 = 0

(x – 4)(x – 8) = 0

∴ x = 4 or 8

∴ y = 8 or 4

Hence, those two numbers are 4 and 8.