A report on consumer financial literacy summarized data from a representative sample of 1,664 adult Americans. Based on data from this sampl

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A report on consumer financial literacy summarized data from a representative sample of 1,664 adult Americans. Based on data from this sample, it was reported that over half of U.S. adults would give themselves a grade of A or B on their knowledge of personal finance. This statement was based on observing that 939 people in the sample would have given themselves a grade of A or B.
(a) Construct and interpret a 95% confidence interval for the proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance. (Round your answers to three decimal places.)
(,)
(b) Is the confidence interval from part (a) consistent with the statement that a majority of adult Americans would give themselves a grade of A or B? Explain why or why not.
Because this confidence interval _____ is entirely below 0.5contains 0.5is entirely above 0.5 , the interval ______ is not consistent with the statement that a majority of adult Americans would give themselves a grade of A or B.

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Calantha 2 days 2021-07-22T08:31:58+00:00 1 Answers 1 views 0

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    2021-07-22T08:32:58+00:00

    Answer:

    a. The 95% confidence interval for the proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is (0.54, 0.588). This means that we are 95% sure that the true proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is between these two bounds.

    b. Because this confidence interval is entirely above 0.5, the interval is consistent with the statement that a majority of adult Americans would give themselves a grade of A or B.

    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}.

    Sample of 1,664 adult Americans, 939 people in the sample would have given themselves a grade of A or B in personal finance.

    This means that n = 1664, \pi = \frac{939}{1664} = 0.5643

    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.

    The lower limit of this interval is:

    \pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5643 - 1.96\sqrt{\frac{0.5643*0.4357}{1644}} = 0.54

    The upper limit of this interval is:

    \pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.5643 + 1.96\sqrt{\frac{0.5643*0.4357}{1644}} = 0.588

    The 95% confidence interval for the proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is (0.54, 0.588). This means that we are 95% sure that the true proportion of all adult Americans who would give themselves a grade of A or B on their financial knowledge of personal finance is between these two bounds.

    (b) Is the confidence interval from part (a) consistent with the statement that a majority of adult Americans would give themselves a grade of A or B?

    Yes, because the confidence interval is entirely above 0.5.

    Because this confidence interval is entirely above 0.5, the interval is consistent with the statement that a majority of adult Americans would give themselves a grade of A or B.

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