Statistics
Statistics
nta_abhyas_2025
Grade None
Question:
If $x_1, x_2, x_3, x_4, x_5, \ldots, x_n$ are $n$ observations such that $\sum_{i=1}^{n} x_i^2 = 400$ and $\sum_{i=1}^{n} x_i = 100$, then the possible value of $n$ among the following is
Step-by-Step Solution
Key Concept: Variance formula $\text{Var}(x) = E(x^2) - [E(x)]^2$ combined with the constraint that variance cannot be negative
We know that $\text{Var}(x) = E(x^2) - [E(x)]^2$. Given $\text{Var}(x) = 16$, we have $E(x^2) - [E(x)]^2 = 16$. We also know that $\sum x^2 - 100n \geq 0$ and $\frac{\text{Var}(x)}{n} = \frac{16}{n}$. Solving the inequality $\frac{480 - 1600}{n} \geq 0$ gives $n \geq 25$. Therefore, the minimum value of $n$ is $27$.
Correct Answer: 27