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(B) Let the two data sets be $X$ and $Y$ with sizes $n_1 = 5$ and $n_2 = 5$. Given: $\sigma_1^2 = 4$,$\sigma_2^2 = 5$,$\bar{x}_1 = 2$,$\bar{x}_2 = 4$.

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$\frac{11}{2}$

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(B) Let the two data sets be $X$ and $Y$ with sizes $n_1 = 5$ and $n_2 = 5$. Given: $\sigma_1^2 = 4$,$\sigma_2^2 = 5$,$\bar{x}_1 = 2$,$\bar{x}_2 = 4$. For set $X$: $\sigma_1^2 = \frac{\Sigma x_i^2}{n_1} - \bar{x}_1^2 \implies 4 = \frac{\Sigma x_i^2}{5} - 4 \implies \Sigma x_i^2 = 40$. For set $Y$: $\sigma_2^2 = \frac{\Sigma y_i^2}{n_2} - \bar{x}_2^2 \implies 5 = \frac{\Sigma y_i^2}{5} - 16 \implies \Sigma y_i^2 = 105$. The combined mean $\bar{x}_{comb} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2} = \frac{5(2) + 5(4)}{10} = 3$. The combined variance $\sigma^2 = \frac{\Sigma x_i^2 + \Sigma y_i^2}{n_1 + n_2} - (\bar{x}_{comb})^2$. $\sigma^2 = \frac{40 + 105}{10} - (3)^2 = \frac{145}{10} - 9 = 14.5 - 9 = 5.5 = \frac{11}{2}$.

Daily wage (Rs.)	$30$-$40$	$40$-$50$	$50$-$60$	$60$-$70$	$70$-$80$	$80$-$90$
No. of workers	$17$	$28$	$21$	$15$	$13$	$6$

Two data sets each of size $5$ are given. If the variances are $4$ and $5$ and their corresponding means are $2$ and $4$ respectively,then what is the variance of the combined data set?

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