A sampling distribution used to estimate a population proportion, p, yielded a value of 0.25. Which of the numbered choices is the margin of

Question

A sampling distribution used to estimate a population proportion, p, yielded a value of 0.25. Which of the numbered choices is the margin of error that corresponds to a sample size of 600 and a confidence level of 95%?

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Athena 8 hours 2021-10-14T12:36:22+00:00 1 Answer 0

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    2021-10-14T12:38:05+00:00

    Answer:

    In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by \alpha=1-0.95=0.05 and \alpha/2 =0.025. And the critical value would be given by:

    z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96

    The margin of error for the proportion interval is given by this formula:  

     ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}    (a)  

    And replacing we got:

     ME= 1.96 \sqrt{\frac{0.25 (1-0.25)}{600}}= 0.0346

    Step-by-step explanation:

    Previous concepts

    A confidence interval is “a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval”.  

    The margin of error is the range of values below and above the sample statistic in a confidence interval.  

    Normal distribution, is a “probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean”.  

    The population proportion have the following distribution

    p \sim N(p,\sqrt{\frac{p(1-p)}{n}})

    Solution to the problem

    In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by \alpha=1-0.95=0.05 and \alpha/2 =0.025. And the critical value would be given by:

    z_{\alpha/2}=-1.96, z_{1-\alpha/2}=1.96

    The margin of error for the proportion interval is given by this formula:  

     ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}    (a)  

    And replacing we got:

     ME= 1.96 \sqrt{\frac{0.25 (1-0.25)}{600}}= 0.0346

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