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)

${\sigma }_1^2=\frac{\sum x_i^2}{100}-40^2$
${\sigma }_2^2=\frac{\sum x_j^2}{100}-55^2$
$350={\sigma }^2=\frac{\sum x_i^2+\sum x_j^2}{300}-(\overline{x}{)}^2$
$350=\frac{\left(1600+{\alpha }^2\right)\times 100+\left[(30-\alpha {)}^2+3025\right]\times 200}{300}-(50{)}^2$
$2850\times 3={\alpha }^2+2(30-\alpha {)}^2+1600+6050$
$8550={\alpha }^2+2(30-\alpha {)}^2+7650$
${\alpha }^2+2(30-\alpha {)}^2=900$
${\alpha }^2-40\alpha +300=0$
$\alpha =10,30$
${\sigma }_1^2+{\sigma }_2^2=10^2+20^2=500$