Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# The mean and variance of 8 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. - Mathematics

The mean and variance of 8 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.

#### Solution

Let x and y be the remaining two observations.

$n = 8$

$\text{ Variance } = 9 . 25$

$\bar{X} = \text{ Mean } = 9$

$\Rightarrow \frac{6 + 7 + 10 + 12 + 12 + 13 + x + y}{8} = 9$

$\Rightarrow 60 + x + y = 72$

$\Rightarrow x + y = 12 . . . (1)$

$\text{ Variance } X = \frac{1}{n} \sum^8_{i = 1} {x_i}^2 - \left( \bar{X} \right)^2$

$\Rightarrow 9 . 25 = \left( \frac{1}{8} \times \left( 6^2 + 7^2 + {10}^2 + {12}^2 + {12}^2 + {13}^2 + x^2 + y^2 \right) \right) - 9^2$

$\Rightarrow 9 . 25 = \frac{1}{8}\left( 642 + x^2 + y^2 \right) - 81$

$\Rightarrow 9 . 25 \times 8 = 642 + x^2 + y^2 - 648$

$\Rightarrow x^2 + y^2 = 80 . . . . (2)$

$\text{ We know } ,$

$\left( x + y \right)^2 + \left( x - y \right)^2 = 2\left( x^2 + y^2 \right)$

$\Rightarrow {12}^2 + \left( x - y \right)^2 = 2 \times 80 \left[ \text{ using equations (1) and (2) } \right]$

$\Rightarrow 144 + \left( x - y \right)^2 = 160$

$\Rightarrow \left( x - y \right)^2 = 16$

$\Rightarrow x - y = \pm 4$

$\text{ If x - y = 4, then x + y = 12 and x - y = 4 give x = 8 and } y = 4$

$\text{ If x - y = - 4, then x + y = 12 and x - y = 4 give x = 4 and } y = 8$

Thus, the remaining two observations are 8 and 4.

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Exercise 32.4 | Q 6 | Page 28