Consider flipping a (biased) coin for which the probability of head is p. The fraction of heads after n independent tosses is Xn. (a) Suppose p = 0.6. Using the Central Limit Theorem, how large should n be

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.2: Arithmetic Sequences
Problem 68E
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Question 3
Consider flipping a (biased) coin for which the probability of head is p. The fraction of heads
after n independent tosses is Xn.
(a)
so that approximately P(0.5 < Xn < 0.7) > 0.95?
Suppose p = 0.6. Using the Central Limit Theorem, how large should n be
Hint: If Z ~
N(0, 1), then P(-1.96 < Z < 1.96) = 0.95.
Note: Compare your result here with Question 4(b) in Homework 1, where you calculated
such n using Chebyshev's inequality. Are the n's you obtained the same/different? Which n
is greater? How would you understand/interpret this difference? (Something to think about;
No need to answer this in your solution.)
Transcribed Image Text:Question 3 Consider flipping a (biased) coin for which the probability of head is p. The fraction of heads after n independent tosses is Xn. (a) so that approximately P(0.5 < Xn < 0.7) > 0.95? Suppose p = 0.6. Using the Central Limit Theorem, how large should n be Hint: If Z ~ N(0, 1), then P(-1.96 < Z < 1.96) = 0.95. Note: Compare your result here with Question 4(b) in Homework 1, where you calculated such n using Chebyshev's inequality. Are the n's you obtained the same/different? Which n is greater? How would you understand/interpret this difference? (Something to think about; No need to answer this in your solution.)
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