6.Suppose the Gallup Organization wants to estimate the population proportion of those who think there should be a law that would ban the po

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6.Suppose the Gallup Organization wants to estimate the population proportion of those who think there should be a law that would ban the possession of handguns. In a previous study of 1012 randomly chosen respondents, 374 said that there should be such a law. How large a sample size is needed to be 95% confident with a margin of error of E

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Kim Cúc 4 years 2021-08-21T21:02:31+00:00 1 Answers 8 views 0

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  1. thienthanh
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    2021-08-21T21:04:14+00:00
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    Answer:

    A sample of n = (\frac{1.96\sqrt{0.3696*0.6304}}{E})^2 is needed, in which E is the desired margin of error, as a proportion. If we find a decimal value, we round up to the next whole number.

    Step-by-step explanation:

    In a sample with a number n of people surveyed with a probability of a success of \pi, and a confidence level of 1-\alpha, we have the following confidence interval of proportions.

    \pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}

    In which

    z is the zscore that has a pvalue of 1 - \frac{\alpha}{2}.

    The margin of error is of:

    M = z\sqrt{\frac{\pi(1-\pi)}{n}}

    In a previous study of 1012 randomly chosen respondents, 374 said that there should be such a law.

    This means that n = 1012, \pi = \frac{374}{1012} = 0.3696

    95% confidence level

    So \alpha = 0.05, z is the value of Z that has a pvalue of 1 - \frac{0.05}{2} = 0.975, so Z = 1.96.

    How large a sample size is needed to be 95% confident with a margin of error of E?

    A sample size of n is needed, and n is found when M = E.

    M = z\sqrt{\frac{\pi(1-\pi)}{n}}

    E = 1.96\sqrt{\frac{0.3696*0.6304}{n}}

    E\sqrt{n} = 1.96\sqrt{0.3696*0.6304}

    \sqrt{n} = \frac{1.96\sqrt{0.3696*0.6304}}{E}

    (\sqrt{n})^2 = (\frac{1.96\sqrt{0.3696*0.6304}}{E})^2

    n = (\frac{1.96\sqrt{0.3696*0.6304}}{E})^2

    A sample of n = (\frac{1.96\sqrt{0.3696*0.6304}}{E})^2 is needed, in which E is the desired margin of error, as a proportion. If we find a decimal value, we round up to the next whole number.

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