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Chebyshev's theorem problem

  1. Sep 27, 2015 #1
    1. The problem statement, all variables and given/known data
    How many times do we have to flip a balanced coin to be able to assert with a probability of at most .01 that the difference between the proportion of tails and .50 will be at least .04?

    2. Relevant equations
    P( |X-μ| ≥ kσ ) ≤ 1/k^2

    3. The attempt at a solution

    I am very confused about how to use this theorem. So far I have only managed to figure out bits and pieces.
    I know that
    P ≤ .01 = 1/k^2 so
    k^2 = 1/.01 = 100
    k = 10

    also μ = np and since the coin is balanced, p = 1/2 so
    μ = n/2
    also σ^2 = np(1-p) = n/2(1/2) = n/4
    so σ = sqrt(n)/2

    plugging this all into the inequality I get

    P( |X-n/2| ≥ (10)(sqrt(n)/2) ) ≤ .01

    P( |X-n/2| ≥ (5)sqrt(n) ) ≤ .01

    But I am still confused about what this means or how I can solve for n (which is what I think I need to be solving for.) please help :(
     
  2. jcsd
  3. Sep 27, 2015 #2
    The random variable X is the proportion of flips that are tails, not the number of tails. Thus mu= 0.5 not n/2 and sigma = sqrt (p(1-p)/n). Try again using these.
     
  4. Sep 27, 2015 #3
    Now things are starting to make sense, thank you so much.
    so
    μ = 1/2
    and σ = 1/sqrt(2n)

    The inequality becomes
    P( |X-1/2| ≥ 10/sqrt(2n) ) ≤ .01
    so

    10/sqrt(2n) = .04

    10/.04 = sqrt(2n)

    250 = sqrt(2n)

    62500 = 2n

    n = 31250
     
  5. Sep 27, 2015 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    It is just as easy to continue using X = number of tails, but to write the desired condition correctly. You want to know the probability of the event
    [tex] \left\{ \left| \frac{X}{n} - \frac{1}{2} \right| > .04 \right\} [/tex]
    Using ##\mu = n/2, \sigma = (1/2) \sqrt{n}## we can re-write the event above is the same as
    [tex] \{ |X - \mu| > 0.04\, n \} = \{ |X - \mu| > 0.08 \sqrt{n} \; (1/2) \sqrt{n} \} = \{ |X - \mu| > 0.08 \sqrt{n} \: \sigma \} [/tex]
     
    Last edited: Sep 27, 2015
  6. Sep 27, 2015 #5
    oops i think i made a mistake.

    would it be σ = 1/(2sqrt(n))

    instead of

    σ = 1/sqrt(2n) ?
     
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