Question

Let the mean and standard deviation of marks of class A of $100$ students be respectively $40$ and $\alpha (> 0 )$, and the mean and standard deviation of marks of class B of $n$ students be respectively $55$ and 30 $-\alpha$. If the mean and variance of the marks of the combined class of $100+ n$ students are respectively $50$ and $350$ , then the sum of variances of classes $A$ and $B$ is :

• A. 500
• B. 450
• C. 650
• D. 900

Answer 1

Answer: (A)

$ \sigma_1^2=\frac{\sum x _{ i }^2}{100}-40^2 $
$ \sigma_2^2=\frac{\sum x _{ j }^2}{100}-55^2 $
$ 350=\sigma^2=\frac{\sum x _{ i }^2+\sum x _{ j }^2}{300}-(\overline{ x })^2$
$350=\frac{\left(1600+\alpha^2\right) \times 100+\left[(30-\alpha)^2+3025\right] \times 200}{300}-(50)^2$
$ 2850 \times 3=\alpha^2+2(30-\alpha)^2+1600+6050 $
$ 8550=\alpha^2+2(30-\alpha)^2+7650 $
$ \alpha^2+2(30-\alpha)^2=900 $
$ \alpha^2-40 \alpha+300=0 $
$ \alpha=10,30 $
$ \sigma_1^2+\sigma_2^2=10^2+20^2=500$