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Questions about Chebyshev's Inequality

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  • Questions about Chebyshev's Inequality

    The scores in a national test have a mean of 200 and standard deviation of 10.
    The scores are continuous and distributed symmetrically about the mean. Of
    the following numbers which is the largest possible value for the probability
    that the score of a randomly chosen candidate is at least 215?
    (A) 0.214 (B) 0.216 (C) 0.218 (D) 0.220 (E) 0.224

    Let X be the score. Since the distribution of the score is symmetric about
    the mean, using Chebyshev’s inequality,
    Pr(X>215) = (1/2)Pr[|X − 200|>15]<(1/2)(10/15)^2 = 2/9.
    The largest of the given numbers that satisfies this is 0.22.

    My question is why we need to multiply 1/2 ?? Is something related to "The scores are continuous and distributed symmetrically about the mean"??

    Please help me!! Thank you very much!!

    10 more days.........
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    I promise you I will learn from my mistakes

  • #2
    Hi Double T,

    Chebyshev's formula is for the probability that the score will be within 'k' standard deviations of the mean...this is indicated by the absolute value of X-mean . Therefore if we only want the probability where X is atleast 215, we have to divide this by 2. Of course, this is possible because it is, as you said, a symmetric distribution about the mean.

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