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Math Help - Probabilty of smoking

  1. #1
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    Probabilty of smoking

    A fraction p of citizens in a city smoke. We are to determine the value of p by making a survey that involves n citizens whom we select randomly. If k of these n people smoke, then p'=k/n will be our result. How large should we choose n if we want our result p' to be closer to the real value p than 0.005 with probability at least 0.95? In other words: determine the smallest number n_0 such that:
    P(|p'-p|)\leq 0.005)\geq 0.95, for any p \in (0,1) and n\geq n_0.

    Thank you very much in advance!
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  2. #2
    Grand Panjandrum
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    Re: Probabilty of smoking

    Quote Originally Posted by zadir View Post
    A fraction p of citizens in a city smoke. We are to determine the value of p by making a survey that involves n citizens whom we select randomly. If k of these n people smoke, then p'=k/n will be our result. How large should we choose n if we want our result p' to be closer to the real value p than 0.005 with probability at least 0.95? In other words: determine the smallest number n_0 such that:
    P(|p'-p|)\leq 0.005)\geq 0.95, for any p \in (0,1) and n\geq n_0.

    Thank you very much in advance!
    We use the normal approximation, so under this approximation we have:

     (p'-p) \sim N(0,p(1-p)/n)

    Now the z-score corresponding to (p'-p)=0.005 is:

    z=\frac{0.005 n}{p(1-p)}

    and the critical value for a 2-sided 95% interval is 1.96 so we find the largest n such that:

    \frac{0.005 n}{p(1-p)}\le 1.96

    for any p \in [0,1] (which will occur when p=0.5)

    CB
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