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Fact: For any normal distribution, 99.7 % of the total area is between three standard deviations to the left of the mean and three standard deviation to the right of the mean. In other words, due to the symmetry, half of that area, or 49.85 % of the total area is between the mean and three standard deviations from the mean.
Fact: A version of Fact 2.5 for the standard normal distribution.
For the standard normal distribution: 99.7 % of the total area is between z = -3 and z = +3. In other words 49.85 % of the total area is between the µ = 0 and z = 3.
Questions:
(a) How much remaining percentage of the total area is unaccounted for?
(b) From what you just computed, compare the remaining percentage to the percentage from Chebyshev's theorem for the |z| > 3. Which percentage is larger? Does the remaining percentage of the total area from (a) violate Chebyshev's theorem?
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