Let x1,x2 ldots ,xn be n observations such that displaystyle ∑ xi2 = 300 and displaystyle ∑ xi = 90. Then a possible value of n among the following is

Question

Let x1,x2 ldots ,xn be n observations such that displaystyle ∑ xi2 = 300 and displaystyle ∑ xi = 90. Then a possible value of n among the following is (A) 25 (B) 18 (C) 29 (D) 22

Tardigrade Admin · Accepted Answer

Variance =( Sigma xi2/n)-(( Sigma xi/n))2 ⇒ ( Sigma xi2/n) ≥(( Sigma xi/n))2 ⇒ n ≥ (( Sigma xi)2/ Sigma xi2) ⇒ n ≥ (8100/300) ⇒ n ≥ 27

Q. Let $x_{1}, x_{2} \dots, x_{n}$ be $n$ observations such that $\sum x_{i}^{2} = 300$ and $\sum x_{i} = 90.$ Then a possible value of $n$ among the following is

$25$

$18$

$29$

$22$

Variance $= \frac{Σ x _{i}^{2}}{n} - (\frac{Σ x _{i}}{n})^{2}$
$\Rightarrow \frac{Σ x _{i}^{2}}{n} \geq (\frac{Σ x _{i}}{n})^{2}$
$\Rightarrow n \geq \frac{( Σ x _{i} ) ^{2}}{Σ x _{i}^{2}} \Rightarrow n \geq \frac{8100}{300} \Rightarrow n \geq 27$

Q. Let x1​,x2​…,xn​ be n observations such that ∑xi2​=300 and ∑xi​=90. Then a possible value of n among the following is

25

18

29

22