If the mean and variance of five observations | vedclass

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Question

(C) Let the five observations be $x_1, x_2, x_3, x_4, x_5$. Given mean $\bar{X} = \frac{24}{5}$ and variance $\sigma^2 = \frac{194}{25}$. Sum of five

Vedclass · Accepted Answer

$\frac{5}{4}$

Vedclass · Answer

(C) Let the five observations be $x_1, x_2, x_3, x_4, x_5$. Given mean $\bar{X} = \frac{24}{5}$ and variance $\sigma^2 = \frac{194}{25}$. Sum of five observations: $\sum_{i=1}^5 x_i = 5 	imes \frac{24}{5} = 24$. Mean of first four observations is $\frac{7}{2}$,so $\sum_{i=1}^4 x_i = 4 	imes \frac{7}{2} = 14$. Thus,$x_5 = 24 - 14 = 10$. Using the variance formula $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{X})^2$: $\frac{194}{25} = \frac{\sum_{i=1}^5 x_i^2}{5} - (\frac{24}{5})^2$. $\frac{194}{25} = \frac{\sum_{i=1}^5 x_i^2}{5} - \frac{576}{25}$ $\Rightarrow \frac{\sum_{i=1}^5 x_i^2}{5} = \frac{770}{25} = \frac{154}{5}$. $\sum_{i=1}^5 x_i^2 = 154$. Since $x_5 = 10$,$x_5^2 = 100$. $\sum_{i=1}^4 x_i^2 = 154 - 100 = 54$. Variance of first four observations: $	ext{Var} = \frac{\sum_{i=1}^4 x_i^2}{4} - (	ext{mean of first four})^2$. $	ext{Var} = \frac{54}{4} - (\frac{7}{2})^2 = \frac{27}{2} - \frac{49}{4} = \frac{54 - 49}{4} = \frac{5}{4}$.

Classes	$30-40$	$40-50$	$50-60$	$60-70$	$70-80$	$80-90$	$90-100$
Frequency $({f_i})$	$3$	$7$	$12$	$15$	$8$	$3$	$2$

\text{Particulars}	\text{Firm } $A$ \text{ and Firm } $B$
\text{No. of wage earners}	$A: 586, B: 648$
\text{Mean of monthly wages}	$A: Rs. 5253, B: Rs. 5253$
\text{Variance of wages}	$A: 100, B: 121$

If the mean and variance of five observations are $\frac{24}{5}$ and $\frac{194}{25}$ respectively,and the mean of the first four observations is $\frac{7}{2}$,then the variance of the first four observations is equal to:

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