Multiple choice

The mean of 6 distinct observations is 6.5 and their variance is 10.25. If 4 out of 6 observations are 2, 4, 5 and 7, then the remaining two observations are:

  1. 10, 11

  2. 8, 13

  3. 1, 20

  4. 3, 18
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A Correct answer
Explanation

Mean = 6.5, Sum = 39. Sum of 4 observations = 18. Remaining two sum to 21. Variance = 10.25 = (Sum(x^2)/6) - (6.5)^2. Sum(x^2) = 6 * (10.25 + 42.25) = 315. Sum of squares of 4 observations = 4+16+25+49 = 94. Remaining two squares sum to 221. x+y=21, x^2+y^2=221. Solving gives 10 and 11.

AI explanation

Since the mean of the six observations is 6.5, their total sum is 39, making the sum of the remaining two observations equal to 21. Using the variance formula, the sum of the squares of all six observations is 6 multiplied by 10.25 plus 39 squared, which equals 555; subtracting the squares of the four known observations leaves 221 for the sum of the squares of the two unknowns. Checking the option 10 and 11, their sum is 21 and the sum of their squares is 221, so the remaining two observations are 10 and 11.

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