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(C) Given the frequency distribution table,the total frequency $N = \sum f_i = 4 + 4 + \alpha + 15 + 8 + \beta + 4 + 5 = 40 + \alpha + \beta$. The sum

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$25$

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(C) Given the frequency distribution table,the total frequency $N = \sum f_i = 4 + 4 + \alpha + 15 + 8 + \beta + 4 + 5 = 40 + \alpha + \beta$. The sum $\sum f_i x_i = (2 	imes 4) + (4 	imes 4) + (6 	imes \alpha) + (8 	imes 15) + (10 	imes 8) + (12 	imes \beta) + (14 	imes 4) + (16 	imes 5) = 8 + 16 + 6\alpha + 120 + 80 + 12\beta + 56 + 80 = 360 + 6\alpha + 12\beta$. The mean $\bar{x} = \frac{\sum f_i x_i}{N} = 9$ $\Rightarrow 360 + 6\alpha + 12\beta = 9(40 + \alpha + \beta)$ $\Rightarrow 360 + 6\alpha + 12\beta = 360 + 9\alpha + 9\beta$ $\Rightarrow 3\beta = 3\alpha$ $\Rightarrow \alpha = \beta$. Substituting $\alpha = \beta$ into $N$,we get $N = 40 + 2\alpha$. The sum $\sum f_i x_i^2 = (4 	imes 4) + (16 	imes 4) + (36 	imes \alpha) + (64 	imes 15) + (100 	imes 8) + (144 	imes \alpha) + (196 	imes 4) + (256 	imes 5) = 16 + 64 + 36\alpha + 960 + 800 + 144\alpha + 784 + 1280 = 3904 + 180\alpha$. Variance $\sigma^2 = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2 = 15.08$ $\Rightarrow \frac{3904 + 180\alpha}{40 + 2\alpha} - 81 = 15.08$ $\Rightarrow \frac{3904 + 180\alpha}{40 + 2\alpha} = 96.08$. $3904 + 180\alpha = 96.08(40 + 2\alpha) = 3843.2 + 192.16\alpha$ $\Rightarrow 12.16\alpha = 60.8$ $\Rightarrow \alpha = 5$. Since $\alpha = \beta$,then $\beta = 5$. The value of $\alpha^2 + \beta^2 - \alpha\beta = 5^2 + 5^2 - (5 	imes 5) = 25 + 25 - 25 = 25$.

$x_i$	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$
$f_i$	$4$	$4$	$\alpha$	$15$	$8$	$\beta$	$4$	$5$

If the mean and variance of the frequency distribution are $9$ and $15.08$ respectively,then the value of $\alpha^2+\beta^2-\alpha \beta$ is $............$.

$x_i$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$

$f_i$ $4$ $4$ $\alpha$ $15$ $8$ $\beta$ $4$ $5$

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If the mean and variance of the frequency distribution are $9$ and $15.08$ respectively,then the value of $\alpha^2+\beta^2-\alpha \beta$ is $............$. $x_i$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $f_i$ $4$ $4$ $\alpha$ $15$ $8$ $\beta$ $4$ $5$