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AIEEE 2010 Solved Paper

Section: Mathematics

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Question : 89 of 89

Marks: +1, -0

For two data sets, each of size 5 , the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4 , respectively. The variance of the combined data set is

$rac{5}{2}$
$rac{11}{2}$
$6$
$rac{13}{2}$

Solution:

Given that,

\sigma_1^2=4

and

\sigma_2^2=5

And also given,

\overline{x}=2

and

\overline{y}=4

So,

rac{\sum x_i}{5}=2

\Rightarrow \sum x_i=10 v

and

rac{\sum y_i}{5}=4

\Rightarrow \sum y_i=20

\therefore \sigma_1^2=\frac{\sum x_i^2}{5}-(\overline{x})^2

\Rightarrow 4=\frac{\sum x_i^2}{5}-4

\Rightarrow \sum x_i^2=40

and

\sigma_2^2=\frac{\sum y_i^2}{5}-(\overline{y})^2

\Rightarrow 5=\frac{\sum y_i^2}{5}-16

\Rightarrow \sum y_i^2=105

Variance of combined data set

\sigma^2=\frac{1}{10}\left(\sum x_i^2+\sum y_i^2\right)-\left(\frac{\overline{x}+\overline{y}}{2}\right)^2

=\frac{1}{10}(40+105)-9

=\frac{145-90}{10}

=\frac{55}{10}

=\frac{11}{2}

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