A researcher would like to estimate p, the proportion of U.S. adults who support recognizing civil unions between gay or lesbian couples.

Question

A researcher would like to estimate p, the proportion of U.S. adults who support recognizing civil unions between gay or lesbian couples.
If the researcher would like to be 95% sure that the obtained sample proportion would be within 1.5% of p (the proportion in the entire population of U.S. adults), what sample size should be used?
(a) 17,778
(b) 4,445
(c) 1,112
(d) 67
(e) 45
Due to a limited budget, the researcher obtained opinions from a random sample of only 2,222 U.S. adults. With this sample size, the researcher can be 95% confident that the obtained sample proportion will differ from the true proportion (p) by no more than (answers are rounded):
(a) .04%
(b) .75%
(c) 2.1%
(d) 3%
(e) There is no way to figure this out without knowing the actual sample proportion that was obtained.

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Kim Chi 3 years 2021-08-06T21:57:33+00:00 1 Answers 28 views 0

Answers ( )

  1. Delwyn
    0
    2021-08-06T21:59:06+00:00
    Reply

    Answer:

    Question 1:

    (b) 4,445

    Question 2:

    (c) 2.1%

    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}}

    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.

    Question 1:

    We have no previous estimate for the population proportion, so we use \pi = 0.5.

    The sample size is n for which M = 0.015. So

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

    0.015 = 1.96\sqrt{\frac{0.5*0.5}{n}}

    0.015\sqrt{n} = 1.96*0.5

    \sqrt{n} = \frac{1.96*0.5}{0.015}

    (\sqrt{n})^2 = (\frac{1.96*0.5}{0.015})^2

    n = 4268

    Samples above this value should be used, and the smaller sample above this value is of 4445, so the answer is given by option b.

    Question 2:

    Now we find M for which n = 2222.

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

    M = 1.96\sqrt{\frac{0.5*0.5}{2222}}

    M = 0.021

    So 2.1%, and the correct answer is given by option c.

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