Question

The mean of five observations is 4.4 and their variance is 8.24. If three of the five observations are 1, 4 and 9, then the product of other two observations is:

• A. 8
• B. 10
• C. 12
• D. 15

Hint:

Use mean and variance definitions to form equations, then solve via identities or substitution.

Answer 1

The Correct Option is C

Let the five numbers be: 1, 4, 9, \( x \), \( y \) Mean = 4.4 ⇒ \[ \frac{1 + 4 + 9 + x + y}{5} = 4.4 \Rightarrow x + y = 22 - 14 = 8 \quad \text{(i)} \] Variance = 8.24 ⇒ \[ \text{Variance} = \frac{1}{5} \left[ (1 - 4.4)^2 + (4 - 4.4)^2 + (9 - 4.4)^2 + (x - 4.4)^2 + (y - 4.4)^2 \right] = 8.24
\Rightarrow 11.56 + (x - 4.4)^2 + (y - 4.4)^2 = 41.2
\Rightarrow (x - 4.4)^2 + (y - 4.4)^2 = 29.64 \quad \text{(ii)} \] Now solve (i) and (ii): use identity \[ (x - a)^2 + (y - a)^2 = (x + y - 2a)^2/2 + (x - y)^2/2 \Rightarrow \text{Get} \, xy = 12 \] Correct value: 12