Question

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

Answer 1

Let those two numbers be x and y.

`overline x = 8 = (2 + 4 + 10 + 12 + 14 + x + y)/7`

or 56 = 42 + x + y or x + y = 56 − 42 = 14       ...(i)

σ² = `1/n^2 [nsumx_i^2 - (sumx_i)^2]`

`[overline x = (sumx_i)/n     ∴ sumx_i = n overline x = 7 xx 8 = 56]`

`σ^2 = 16 = 1/49 [7 xx sumx_i^2 - (56)^2]`

∴ `7sumx_i^2` = 49 × 16 + 56 × 56

or `sumx_i^2` = 7 × 16 + 8 × 56

= 560

or 2² + 4² + 10² + 12² + 14² + x² + y²

= 560

460 + x² + y² = 560

x² + y² = 560 – 460 = 100      ...(ii)

From equations (i) and (ii),

x² + (14 – x)² = 100

or 2x² – 28x + 196 – 100 = 0

or x² – 14x + 48 = 0

∴ (x – 6)(x – 8) = 0

∴ x = 6 or 8

∴ y = 8 or 6

∴ Those two numbers are 6 and 8.