Question

Question 1 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.

Class X1 - Maths -Statistics Page 380

Answer 1

Let two remaining observations are P and Q
given , mean = 9 and σ² = 9.25
but , mean = sum of observations /total number of observations
9 = (6 + 7 + 10 + 12 + 12 + 13 + P + Q )/8
9 × 8 = 60 + (P + Q )
72 - 60 = (P + Q)
P + Q = 12 -------------(1)

now,
σ² = 9.25
but σ² = ∑x²/n - (∑x/n)²
9.25 = (6² + 7² + 10² + 12² + 12² + 13² + P² + Q² )/8 - (9)²
9.25 = (642 + P² + Q² )/8 - 81
81 + 9.25 = (642 + P² + Q² )/8
90.25 × 8 = 642 + P² + Q²
722 - 642 = P² + Q²
80 = P² + Q² ------------(2)

from equation (1) and (2),
P² + (12 - P)² = 80
2P² - 24P + 144 - 80 = 0
P² - 12P + 32 = 0
(P - 4)(P - 8) = 0
P = 4, 8
and Q = 8, 4

hence, two remaining observations are 4 and 8