Multiple choice

Let $\mathrm{x}{1},\ \mathrm{x}{2},...........,\ \mathrm{x}{\mathrm{n}}$ be $\mathrm{n}$ observations such that $\displaystyle \sum \mathrm{x}{\mathrm{i}}^{2}=400$ and $\displaystyle \sum \mathrm{x}_{\mathrm{i}}=80$. Then a possible value of $\mathrm{n}$ among the following is

  1. $15$
  2. $18$
  3. $9$
  4. $12$
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B Correct answer
Explanation

Variance = (Sum(xi^2)/n) - (Sum(xi)/n)^2. Since variance must be >= 0, (400/n) - (80/n)^2 >= 0. This simplifies to 400/n >= 6400/n^2, or 1 >= 16/n, so n >= 16. Among the choices, 18 is the only value >= 16.

AI explanation

By the Cauchy-Schwarz inequality, the square of the sum of observations is less than or equal to n times the sum of their squares. Substituting the given values gives 80 squared is less than or equal to n multiplied by 400, which simplifies to 6400 is less than or equal to 400n and n must be at least 16. The value of n must also be strictly less than the sum of the squares since each observation would have to be exactly 1 for n to equal 400, meaning n must be less than 400. Among the choices, only 18 satisfies the condition of being between 16 and 400. Therefore, the possible value is 18.

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