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Thread: Confidence Interval

  1. #1
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    Confidence Interval

    Missed class and have no idea how to start.

    Question is Construct a 90% confidence interval for the proportion of smokers who quit smoking by using either the patch or e-cigarettes within 6 months.
    There was a sample size of 584 and just 38 of the smokers given the e-cigarette or the patch.
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  2. #2
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    Re: Confidence Interval

    I don't think the question is written correctly. Are you saying that there are 584 people who quit smoking, and 38 of those used the e-cigarette or patch?
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    Re: Confidence Interval

    yes i believe thats what they mean. Yeah it isnt written very well.
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    Re: Confidence Interval

    Could this be the equation?

    38/584=0.07
    0.07 (1.645) √(0.07)(0.93)/584
    =0.053,0.087
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    Re: Confidence Interval

    Ok so we can see that $\displaystyle \begin{align*} \mu = \hat{p} = \frac{38}{584} = \frac{19}{292} \end{align*}$ and $\displaystyle \begin{align*} \sigma = \sqrt{\frac{\hat{p} \left( 1 - \hat{p} \right) }{n}} = \sqrt{\frac{\frac{19}{292} \cdot \frac{273}{292}}{584}} = \frac{\sqrt{757\,302}}{85\,264} \end{align*}$. Since we want a 90% Confidence Interval, we want the endpoints which have 90% of the area between them, so we want $\displaystyle \begin{align*} z_{0.05} = -1.645 \end{align*}$ and $\displaystyle \begin{align*} z_{0.95} = 1.645 \end{align*}$. So using the standardisation formula

    $\displaystyle \begin{align*} z &= \frac{x - \mu}{\sigma} \\ -1.645 &= \frac{x - \frac{19}{292}}{\frac{\sqrt{757\,302}}{85\,264}} \\ -1.645 \cdot \frac{\sqrt{757\,302}}{85\,264} &= x - \frac{19}{292} \\ x &= \frac{19}{292} - 1.645 \cdot \frac{\sqrt{757\,302}}{85\,264} \\ x &\approx 0.0483 \end{align*}$

    Now do the same with the other z value.
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    Re: Confidence Interval

    Quote Originally Posted by Simplelife1107 View Post
    Could this be the equation?

    38/584=0.07
    0.07 (1.645) √(0.07)(0.93)/584
    =0.053,0.087
    You appear to have a LOT of roundoff error. You should only ever make approximations where exact values are not possible!
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