Multiple choice

A data consists of $n$ observation: $x_1,x_2,........,x_n.$ If $\displaystyle \sum_{i=1}^{n}(x_i+1)^2=9n$ and $\displaystyle \sum_{i=1}^n(x_i-1)^2=5n$, then the standard deviation of this data is:

  1. $5$
  2. $\sqrt{5}$
  3. $\sqrt{7}$
  4. $2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Variance = (1/n) * sum(x_i^2) - mean^2. Given sum(x_i+1)^2 = 9n and sum(x_i-1)^2 = 5n. Expanding: sum(x_i^2 + 2x_i + 1) = 9n and sum(x_i^2 - 2x_i + 1) = 5n. Adding them: 2*sum(x_i^2) + 2n = 14n => sum(x_i^2) = 6n. Subtracting them: 4*sum(x_i) = 4n => mean = 1. Variance = (6n/n) - 1^2 = 5. Standard deviation = sqrt(5).

AI explanation

Expand the given equations to get the sum of x_i squared plus 2 times the sum of x_i plus n equals 9n, and the sum of x_i squared minus 2 times the sum of x_i plus n equals 5n. Subtracting the second equation from the first yields 4 times the sum of x_i equals 3n, meaning the sum of x_i is 3n/4 and the mean is 3/4. Substituting this back into the first equation shows the sum of x_i squared is 35n/8, making the variance (35n/8 divided by n) minus the square of 3/4, which equals 5. Therefore, the standard deviation is the square root of 5.

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