Mean of five observations is 5 and their variance is 9.20 . If three of given five observations are

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13.Statistics

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The mean of five observations is $5$ and their variance is $9.20$. If three of the given five observations are $1, 3$ and $8$, then a ratio of other two observations is

A

$10 : 3$

B

$4 : 9$

C

$5 : 8$

D

$6 : 7$

(JEE MAIN-2019)

Solution

$\mu = \frac{{1 + 3 + 8 + x + y}}{5}$

$25 = 12 + x + y \Rightarrow x + y = 13\,\,\,\,\,\,\,\,……..\left( 1 \right)$

${\sigma ^2} = \frac{{\sum {{{\left( {{x_i} – \mu } \right)}^2}} }}{N}$

$9.2 = \frac{{1 + 9 + 64 + {x^2} + {y^2}}}{5} – 25$

$34.2 \times 5 = 74 + {x^2} + {y^2}$

$171 = 74 + {x^2} + {y^2}$

$97 = {x^2} + {y^2}……….\left( 2 \right)$

${\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy$

$169 – 97 = 2xy \Rightarrow xy = 36$

$T = 4,9$

So, ratio is $\frac{4}{9}$ or $\frac{9}{4}$

Standard 11

Mathematics

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