The Pew Research Center survey reported that 55% of young singles are unattached

Question

The Pew Research Center survey reported that 55% of young singles are unattached. The size of the random sample was 1080. Reported to the nearest Whole percentages, this survey had a margin of error of 3%, so the 95% confidence interval was 52% to 58%. Thus, based on their survey, The Pew Research Center correctly concludes they are 95% confident that the percentage of young singles who are unattached is between 52% and 58%.” Address parts (a) and (b).

(a) Does this mean 0.95 is the probability that the true, but unknown, percentage of young singles who are unattached is a value between 52% and 58%? Explain.

(b) After checking the requisite conditions (show this), use pˆ ± ME = pˆ ± z*SE = pˆ ± z* pˆqˆ to construct a 99% confidence interval for the percentage of young singles that are unattached. (i) What value must be looked-up in the Table Z (in Appendix F of our textbook) in order to obtain the value of z* required in this problem?

(ii) What is the margin of error of this confidence interval?

Explanation / Answer

sample size = 1080

Pr(Yound singles are unattached) = 0.55

Margin of error = 3%

(a) Here, the given statement "the probability that the true, but unknown, percentage of young singles who are unattached is a value between 52% and 58%" is not true. Here the 95% confidence interval means that if repeated samples of size 1080 will be taken then there is 95% probability that the sample proportion of young singles who are unattached will be in between 52% to 58%.

(b) For 99% confidence interval, the value of Z = 2.575 ; so the value we must look the value 0.995 in the Z table.

(iii) Margin of error of this confidence interval = Z * SE

SE = sqrt [0.55 * 0.45/1080] = 0.01514

Margin of error of this confidence interval = Z * SE = 2.575 * 0.01514 = 0.039

Margin of error = 3.9%

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