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The mean of the numbers $a,b,8,5,10\; is\; 6$ and the variance is $6.80.$ Then which one of the following gives possible values of $a\; \; and\; \; b$ ?

Option 1)
$a=3,b=4$

Option 2)
$a=0,b=7$

Option 3)
$a=5,b=2$

Option 4)
$a=1,b=6$

Answers (1)

best_answer

As we learnt in

ARITHMETIC Mean -

For the values x₁, x₂, ....x_n of the variant x the arithmetic mean is given by

$\bar{x}= \frac{x_{1}+x_{2}+x_{3}+\cdots +x_{n}}{n}$

in case of discrete data.

Variance -

In case of discrete data

$\dpi{100} \sigma ^{2}= \left ( \frac{\sum x_{i}^{2}}{n} \right )-\left ( \frac{\sum x_{i}}{n} \right )^{2}$

mean= $\frac{a+b+8+5+10}{5}=6$

$\Rightarrow a+b=7 - - - - - - \left ( i \right )$

variance = $\frac{\left ( a-b \right )^{2}+\left ( b-b \right )^{2}+\left ( 8-6 \right )^{2}+\left ( 5-6 \right )^{2}+\left ( 10-6 \right )^{2}}{5} =6\cdot 80$

$\Rightarrow a^{2}+b^{2}-12a-12b+36+36+4+1+16=34$

$\Rightarrow a^{2}+b^{2}-12\left ( a+b \right )= -59$

$\Rightarrow a^{2}+b^{2}= -59+12\times 7 \left$ from $\left ( i \right )$

$=25 - - -- - - - -\left ( ii \right )$

Solve (i) and (ii), we get

$\left ( a,b \right )=\left \{ \left ( 3,4 \right )or\left ( 4,3 \right ) \right \}$

Option 1)

$a=3,b=4$

Correct option

Option 2)

$a=0,b=7$

Incorrect option

Option 3)

$a=5,b=2$

Incorrect option

Option 4)

$a=1,b=6$

Incorrect Option

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