In a recent survey, 808 American adults were asked if they would give the U.S. president a favorable job performance rating. Of those surveyed 53% gave the president a favorable rating.Calculate a confidence interval for the true percent of all American adults who would give a favorable rating for the president.Group of answer choices(49.48%, 56.52%)(48.38%, 57.62%)(52.9%, 53.12%)(46.78%, 59.22%)(46.08%, 59.92%)
Question
In a recent survey, 808 American adults were asked if they would give the U.S. president a favorable job performance rating. Of those surveyed 53% gave the president a favorable rating.Calculate a confidence interval for the true percent of all American adults who would give a favorable rating for the president.Group of answer choices(49.48%, 56.52%)(48.38%, 57.62%)(52.9%, 53.12%)(46.78%, 59.22%)(46.08%, 59.92%)
Solution
To calculate the confidence interval, we first need to find the standard error. The formula for standard error (SE) is sqrt[p(1-p)/n], where p is the proportion (in this case, 0.53) and n is the sample size (in this case, 808).
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Calculate the standard error (SE): SE = sqrt[0.53(1-0.53)/808] = 0.017
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Next, we calculate the margin of error (ME). The ME is calculated by multiplying the standard error by the z-score. For a 95% confidence interval, the z-score is 1.96. ME = 1.96 * SE = 1.96 * 0.017 = 0.033
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Finally, we calculate the confidence interval by subtracting the margin of error from the proportion for the lower limit and adding the margin of error to the proportion for the upper limit. Lower limit = 0.53 - 0.033 = 0.497 Upper limit = 0.53 + 0.033 = 0.563
So, the 95% confidence interval for the true percent of all American adults who would give a favorable rating for the president is (49.7%, 56.3%). The closest answer choice to this is (49.48%, 56.52%).
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