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06. Chebyshev theorem - Some examples

# 06. Chebyshev theorem - Some examples - X is not known...

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University of California, Los Angeles Department of Statistics Statistics 100B Instructor: Nicolas Christou Chebyshev’s theorem - some examples Example 1 Let X be a random variable with μ = 11 and σ 2 = 9. Using Chebyshev’s theorem, find the following: a. A lower bound for P (6 < X < 16). [Ans. 0.64] b. The value of c such that P ( | X - 11 | ≥ c ) 0 . 09. [Ans. 10] Example 2 The U.S. mint produces dimes with an average diameter of 0.5 and a standard deviation of 0.01. Using Chebyshev’s theorem, find a lower bound for the number of coins in a lot of 400 coins having diameter between 0.48 and 0.52. [Ans. 300] What would be the exact answer if the diameter is normally distributed with μ = 0 . 5 and σ = 0 . 01? [Ans. 381.76] Example 3 The number of customers per day at a certain sales counter denoted by X , has been observed for a long period of time and found to have a mean of 20 customers and standard deviation of 2 customers. The probability distribution of
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Unformatted text preview: X is not known. Using Chebyshev’s theorem what can be said about the probability that X will be between 16 and 24 customers tomor-row? Example 4 A manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the mileage on tires he sells. Suppose that μ = 25000 and σ = 4000. What interval would you suggest? (Use the Chebyshev’s theorem). [Ans. 12351,37649]. What would be the answer if it is known that the mileage follows the normal distribution? [Ans. 18420,31580] Example 5 A machine used to ﬁll cereal boxes dispenses on average μ ounces per box. The manufacturer wants the actual ounces dispensed, X , to be within 1 ounce of μ at least 75% of the time. What is the largest value of σ that can be tolerated if the manufacturer’s objectives are to be met? (Use Chebyshev’s theorem). [Ans. 1 2 ]....
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