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Math Help - binomial distributions and limit theorems!!!

  1. #1
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    binomial distributions and limit theorems!!!

    find th smallest value of n in a binomial distribution for which we can assert:
    P{ |(Xn/n) p| < .1} ≥ .9
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by lauren2988 View Post
    find th smallest value of n in a binomial distribution for which we can assert:
    P{ |(Xn/n) – p| < .1} ≥ .9
    Well this translates into: find the smallest n such that:

    P(p-0.1<\frac{X_n}{n} <p+0.1) \ge 0.9

    where the rv X_n the number of successes in n independent trials with probability of success in a single trial of p. Now given the title of this thread I assume we are supposed to use a normal approximation to the binomial to do this, in which case we have:

    \frac{X_n}{n} \sim N(p, p(1-p)/n).

    90\% of the probabilty mass of a normal RV is contained within \pm 1.645 standard deviations of the mean, so if p>0.1 we have:

    0.1 \ge 1.645 \sqrt{p(1-p)/n}

    which will allow you to find the smallest n for which this is true.

    Now you can do the case where p<0.1 yourself.

    CB
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