Suppose that a BART ride from Berkeley to San Francisco takes a mean time of 38 minutes with a standard deviation of 4 minutes. If you want to make the claim βAt least 90% of BART rides from Berkeley to San Francisco take between _______ and ______minutes" what numbers should be used to fill in the blanks? I'm lost as to how to manipulate Markov or Chebychev's Theorems to approach this problem. Is there another way to get an answer?
Markov/Chebychev Bound Probability Question
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$\begingroup$
probability
statistics
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0This seems like a straightforward application of Chebyshev's inequality. I put in an answer, but I'd be concerned that this doesn't seem fairly ordinary to you. β 2017-01-31
2 Answers
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You require $~\mathsf P(\mu-x\leqslant X\leqslant \mu+x)~=~0.90~$ for some value, $x$, and the random variable, $X$, whose mean, $\mu$, is $34$ minutes, and standard deviation, $\sigma$, is $4$ minutes. You seek to evaluate $x$.
What kind of distribution do you suppose $X$ might have?
(Hintage: c_ l_ t_)
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0I'm not sure how the CLT helps here, if the distribution is degeneratively trimodal, with an impulse at some length of time less than $38$ minutes, an impulse of magnitude just less than $0.9$ right at $38$ minutes, and another impulse at some length of time greater than $38$ minutes. β 2017-01-31
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Use Chebyshev's Inequality with $k = \sqrt{10}$. This puts all but $1/k^2 = 1/10$ of the distribution inside $k$ standard deviations.