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Suppose the IQ scores of a million individuals have a mean of 100 and an SD of 10.

a)Without making any further assumptions about the distribution of the cores, find an upper bound on the number of scores exceeding 130

b)Find a smaller upper bound on the number of scores exceeding 130 assuming the distribution of scores is symmetric about 100.

For part a. I used Chebychev's Inequality to calculate the upper bound which the probability is $1/3^2 = 1/9$

And for part b, I understand that if the distribution is symmetric about 100 implies that $P(X\ge 130) = P(X\le 70)$, but I am not sure how to get a smaller upper bound.

Can someone help me here? And how many methods of finding an upper bound of a distribution are there in general?

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up vote 5 down vote accepted

Chebyshev’s inequality bounds the probability of being at least $k$ standard deviations from the mean in either direction. If the distribution is symmetric, only half of the outcomes that are at least $k$ sigmas from the mean are on the high side.

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