Multiple choice

The mean and variance of 8 observations are 10 and 13.5, respectively. If 6 of these observations are 5, 7, 10, 12, 14, 15, then the absolute difference of the remaining two observations is:

  1. 9

  2. 3

  3. 7

  4. 5
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C Correct answer
Explanation

Mean = (sum)/8 = 10, so sum = 80. Sum of 6 given = 5+7+10+12+14+15 = 63. Remaining two (x, y) sum to 17. Variance = (sum(xi^2)/8) - mean^2 = 13.5. Sum(xi^2)/8 - 100 = 13.5, so sum(xi^2) = 113.5 * 8 = 908. Sum of squares of 6 given = 25+49+100+144+196+225 = 739. x^2 + y^2 = 908 - 739 = 169. We have x+y=17 and x^2+y^2=169. (x+y)^2 = x^2+y^2+2xy => 289 = 169 + 2xy => 2xy = 120 => xy = 60. Roots of t^2 - 17t + 60 = 0 are 12 and 5. Difference = 12-5 = 7.

AI explanation

The sum of all 8 observations is 8 multiplied by 10, giving a total of 80. Let the remaining two observations be x and y; the sum of the 6 known observations is 5 + 7 + 10 + 12 + 14 + 15 = 63, so x + y = 80 - 63 = 17. Using the variance formula, the sum of the squares of all 8 observations is (variance times total observations) plus (mean squared times total observations), which is (13.5 * 8) + (10^2 * 8) = 108 + 800 = 908. The sum of the squares of the 6 known observations is 25 + 49 + 100 + 144 + 196 + 225 = 739, so x^2 + y^2 = 908 - 739 = 169. Using the identity (x - y)^2 = x^2 + y^2 - 2xy, and knowing 2xy = 2 multiplied by ((17)^2 - 169) / 2 = 120, we find (x - y)^2 = 169 - 120 = 49, making the absolute difference 7.

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