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Statistics Part 2

Section: Mathematics

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Question : 36 of 50

Marks: +1, -0

If the mean and variance of five observations are

\frac{24}{5}

\frac{194}{25}

respectively and the mean of first four observations is

\frac{7}{2}

, then the variance of the first four observations in equal to

[29-Jan-2024 Shift 2]

$\frac{4}{5}$
$\frac{77}{12}$
$\frac{5}{4}$
$\frac{105}{4}$

Solution: 👈: Video Solution

\overline{X}=\frac{24}{5} ; \sigma^2=\frac{194}{25}

Let first four observation be

x_1, x_2, x_3, x_4

Here,

\frac{x_1+x_2+x_3+x_4+x_5}{5}=\frac{24}{5} \dots

(1)

Also,

\frac{x_1+x_2+x_3+x_4}{4}=\frac{7}{2}

\Rightarrow x_1+x_2+x_3+x_4=14

Now from eqn -1

x_5=10

Now,

\sigma^2=\frac{194}{25}

\frac{x_1^2+x_2^2+x_3^2+x_4^2+x_5^2}{5}-\frac{576}{25}

=\frac{194}{25}

\Rightarrow x_1^2+x_2^2+x_3^2+x_4^2=54

Now, variance of first 4 observations

\text{Var} = \frac{\overset{4}{\sum_{i=1} x_i^2}}{4} - \left(\frac{\overset{4}{\sum_{i=1} x_i}}{4}\right)^2

=\frac{54}{4}-\frac{49}{4}=\frac{5}{4}

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