Statistics Question 6
Question 6 - 2024 (29 Jan Shift 2)
If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively and the mean of first four observations is $\frac{7}{2}$, then the variance of the first four observations in equal to
(1) $\frac{4}{5}$
(2) $\frac{77}{12}$
(3) $\frac{5}{4}$
(4) $\frac{105}{4}$
Show Answer
Answer (3)
Solution
$\bar{X}=\frac{24}{5} ; \sigma^{2}=\frac{194}{25}$
Let first four observation be $x _1, x _2, x _3, x _4$
Here, $\frac{x _1+x _2+x _3+x _4+x _5}{5}=\frac{24}{5} \ldots$
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, $\sigma^{2}=\frac{194}{25}$
$$ \begin{aligned} & \frac{x _1^{2}+x _2^{2}+x _3^{2}+x _4^{2}+x _5^{2}}{5}-\frac{576}{25}=\frac{194}{25} \\ & \Rightarrow x _1^{2}+x _2^{2}+x _3^{2}+x _4^{2}=54 \end{aligned} $$
Now, variance of first 4 observations
$$ \begin{aligned} \operatorname{Var} & =\frac{\sum _{i=1}^{4} x _i^{2}}{4}-\left(\frac{\sum _{i=1}^{4} x _i}{4}\right)^{2} \\ & =\frac{54}{4}-\frac{49}{4}=\frac{5}{4} \end{aligned} $$
