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.

Question

Philoid · Accepted Answer

Let us assume the remaining two observations to be x and y respectively such that,

Observations: 6, 7, 10, 12, 12, 13, x, y.

∴ Mean,

60 + x + y = 72

x + y = 12 (i)

By using (i)

So, from equation (i) we have:

Thus, from (ii) and (iii), we have

2xy = 64 (iv)

Now by subtracting (iv) from (ii), we get:

x² + y² – 2xy = 80 – 64

x – y = 4 (v)

Hence, from equation (i) and (v) we have:

When x – y = 4 then x = 8 and y = 4

And, when x – y = - 4 then x = 4 and y = 8

∴ The remaining observations are 4 and 8