Multiple choice

If the mean of a data set a1, a2, ..., a2n is 99, and the sum of the deviations of the data set values measured from 100 is denoted by K, while the sum of the deviations of the values measured from 92 is 140, then what is the value of K?

  1. -30

  2. -20

  3. -10

  4. 10

  5. 20
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B Correct answer
Explanation

Mean = 99. Sum of 2n values = 99 * 2n = 198n. Sum of deviations from 100: sum(a_i - 100) = sum(a_i) - 200n = 198n - 200n = -2n = K. Sum of deviations from 92: sum(a_i - 92) = sum(a_i) - 184n = 198n - 184n = 14n = 140. So n = 10. Thus K = -2(10) = -20.

AI explanation

The sum of deviations measured from a specific value equals the number of observations multiplied by the difference between the actual mean and that specific value. For the deviations from 92, the equation is 2n times (99 minus 92) equals 140, so 2n times 7 equals 140 and 2n equals 20. Using this value of 2n to find K for deviations from 100, we get 20 times (99 minus 100), which equals 20 times negative 1 or negative 20.

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