5. Chebyshev’s Theorem states that for any shape distribution, the minimum propo

Question

5. Chebyshev’s Theorem states that for any shape distribution, the minimum proportion of the distribution’s values will lie within k standard deviations of the mean (for all values of k>1).

a. For a non-normal distribution of weights of 6 week old adorable Labrador Retriever puppies, with a mean of 12 pounds and a standard deviation of 2 pounds, at least what percentage of the puppies will have weights between 8 and 16 pounds? If the distribution of puppy weights were normal, what percentage of the puppies would have weights between 8 and 16 pounds?

b. Explain why the answers to the two questions in part a are entirely consistent with one another.

c. The time it takes for the Far West Shuttle bus to travel from Far West Blvd. to it’s first on-campus stop is 18 minutes on average with a standard deviation of 2 minutes. The distribution is not normal. At least 84% of the trips will take between ______ and ______ minutes.

Explanation / Answer

a)

Here, k = 2 standard deviation from the mean.

By Chebyshev's theorem, at least 1 - 1/2^2 = 0.75 of Labradors are between 8 and 16 pounds.

By normal distirbution's empirical rule, approx 0.95 of all Labradors are within 8 to 16 lbs.

***************

b)

Chebyshev's theorem says "at least 0.75", and 0.95 is more than 0.75. So they are consistent.

***************

1 - 1/k^2 = 0.84

1/k^2 = 0.16

k = 2.5

Thus, 84% of trips lie 2.5 standard deviations from the mean.

Thus,

2.5 s = 2.5(2) = 5.

Thus, 84% lie between 18 - 5 and 18 +5, or 13 to 23 minutes. [ANSWER]

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