Question

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are ______.

Options

• 10, 11

• 8, 13

• 1, 20

• 3, 18

Answer 1

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are 10, 11.

Explanation:

Given, mean `barx = (sumx_i)/n` = 6.5

⇒ `sumx_i` = 6.5 × 6 = 39

Let the remaining two numbers be x and y.

So, 18 + x + y = 39

⇒ x + y = 21  ...(i)

∴ 10.25 = `(sumx_i^2)/n - (barx)^2`

⇒ 10.25 = `(x^2 + y^2 + 4 + 16 + 25 + 49)/6 - (6.5)^2`

⇒ 10.25 = `(x^2 + y^2 + 94)/6 - (6.5)^2`

⇒ x² + y² = 221

Solving (i) and (ii)

⇒ x² + (21 – x)² = 221

⇒ 2x² – 42x + 220 = 0

⇒ x² – 21x + 110 = 0

⇒ (x – 10)(x – 11) = 0

⇒ x = 10, 11

So, x = 10, y = 11