"quantity"exercise: Chebyshev’s inequality

A is correct. According to Chebyshev’s inequality, the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 – 1/k2 for all k >1. For k = 3, that proportion is 1 – 1/32, which is 88.89%. The lower endpoint is, therefore, the mean (382) minus 3 times 14 (the standard deviation), and the upper endpoint is 382 plus 3 times 14 (i.e., 340 and 424, respectively).

C is incorrect; it is based on 2 standard deviations, not 3: 382 – 2 × 14 = 354 and 382 + 2 × 14 = 410. B is incorrect; it centers the interval around the number of observations (438) rather than the mean (382).

B is correct. Technicians believe that security price movements occur before fundamental developments of a company unfold and before they are reported to the public.

A is incorrect. Technicians believe that security price movements occur before, not after, fundamental developments unfold—certainly before they are reported.

C is incorrect. Technicians believe that security price movements occur before, not at the same time as, fundamental developments unfold—certainly before they are reported.