In a national survey, each respondent in a random sample of 1, 500 randomly sele

Question

In a national survey, each respondent in a random sample of 1, 500 randomly selected adults in the U.S. were asked what they thought the level of political division will be in five years. Of those sample, 541 said that the country will be more politically divided a) Calculate an 80 percentage confidence interval for the proportion of all U.S. adults who believe the country will be more politically divided in five years. b) Interpret the confidence interval you calculated in part a) in the context of the problem c) confidence interval you calculated in part a) valid? In other words, are the required conditions met for a valid confidence interval for a population mean? Justify your answer d) what "80 percentage confident" means in the context of this problem. e) Suppose a similar study is performed, but with three times the sample size as the original (i.e., n 4500). The resulting sample proportion is the same in this new study as it was in the original study. How would a 95 percentage confidence interval from this new study compare to the 80 percentage confidence interval you calculated in part a)? You should compare the centers of the two intervals and the widths (or sampling errors) of the two intervals. NOTE: You will need to do some calculations to help you answer this question. Suppose you would like to perform a similar study to estimate the population proportion of U.S. adults who believe the country will be more politically divided in 5 years using a 95 percentage confidence interval. You want your confidence interval to have a margin of error (i.e., sampling error of 0.02. Assuming that 35 percentage of all U.S. adults believe the country will be more politically divided, what sample size is needed? g) Recalculate the sample size in part f assuming you have no knowledge of the true population proportion of U.S. adults who believe the country will be more politically divided in 5 years

Explanation / Answer

Here p = 541/1500 = 0.3607

n = 1500

sigma = sqrt(p*(1-p)/n) = 0.0124

As n is very large here, we will use z-value for the determination of CI

For 80% CI, z-value is +/- 1.28

Lower Limit = p - z*sigma = 0.3607 - 1.28*0.0124 = 0.3448

Upper Limit = p + z*sigma = 0.3607 + 1.28*0.0124 = 0.3766

(A) Hence 80% CI is (0.3448 , 0.3766)

(B) Interpretation: The above calculated 80% CI depicts if we take samples from the population, people who feel the country will be more politically divided is between 34.48% to 37.66%.

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