The mean and variance of seven observations a | vedclass

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(D) Let the $7$ observations be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$.Given mean $\bar{x} = 8$,so $\sum_{i=1}^{7} x_i = 7 	imes 8 = 56$.Given variance

Vedclass · Accepted Answer

$48$

Vedclass · Answer

(D) Let the $7$ observations be $x_1, x_2, x_3, x_4, x_5, x_6, x_7$.Given mean $\bar{x} = 8$,so $\sum_{i=1}^{7} x_i = 7 	imes 8 = 56$.Given variance $\sigma^2 = 16$,we use the formula $\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2$.$16 = \frac{1}{7} \sum_{i=1}^{7} x_i^2 - 8^2$.$16 = \frac{1}{7} \sum_{i=1}^{7} x_i^2 - 64 \Rightarrow \sum_{i=1}^{7} x_i^2 = 7 	imes 80 = 560$.Given $5$ observations are $2, 4, 10, 12, 14$. Let the remaining two be $x_6$ and $x_7$.Sum of $5$ observations: $2 + 4 + 10 + 12 + 14 = 42$.$x_6 + x_7 = 56 - 42 = 14$.Sum of squares of $5$ observations: $2^2 + 4^2 + 10^2 + 12^2 + 14^2 = 4 + 16 + 100 + 144 + 196 = 460$.$x_6^2 + x_7^2 = 560 - 460 = 100$.We know $(x_6 + x_7)^2 = x_6^2 + x_7^2 + 2x_6x_7$.$14^2 = 100 + 2x_6x_7$.$196 = 100 + 2x_6x_7$ $\Rightarrow 2x_6x_7 = 96$ $\Rightarrow x_6x_7 = 48$.

Class	Frequency
$0 - 20$	$17$
$20 - 40$	$f_1$
$40 - 60$	$32$
$60 - 80$	$f_2$
$80 - 100$	$19$
Total	$120$

The mean and variance of seven observations are $8$ and $16$,respectively. If $5$ of the observations are $2, 4, 10, 12, 14$,then the product of the remaining two observations is

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Class Frequency
$0 - 20$ $17$
$20 - 40$ $f_1$
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Total $120$

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