There are $n$ scores $X_1,X_2,X_3,....,X_n$ and their sum is $80$ and sum of their squares is $400$ then which among them is the probable value of $n$
A)$10$
B)$9$
C)$15$
D)$18$
I have tried the following:
$$standard\ deviation\ \sigma=\sqrt{\frac{{\Sigma x_i^2}}{n}-(\bar{x}^2)}$$ $$=\sqrt{\frac{{\Sigma x_i^2}}{n}-(\frac{\Sigma x_i}{n})^2}$$ $$=\sqrt{\frac{400}{n}-\frac{(80)^2}{n^2}}$$ $$=\sqrt{\frac{400}{n}-\frac{6400}{n^2}}$$
But i don't know what to do further.
By the Cauchy-Schwarz Inequality:
$(1^2+1^2+\ldots +1^2)(X_1^2+X_2^2+\ldots +X_n^2)\geq (1*X_1+1*X_2+\ldots +1*X_n)^2$
$n\times 400\geq 80^2$
$n\geq 16$
Therefore the only possible answer is (D).
Alternatively, you could plug in values into the expression under the square root (all values except $18$ give you negative values).