5 In a recent Gallup poll 55 of a sample of 1028 US adults s

5. In a recent Gallup poll, 55% of a sample of 1028 U.S. adults supported same-sex marriage. (a) Treating the data as a simple random sample, find a 95% confidence interval for the percentage of all U.S. adults who support same-sex marriage. (b) (1 point of extra credit.) In fact, Gallup stated a 4 percentage point margin of error at the 95% confidence level, implying a confidence interval of 51% to 59%. Why does this differ from your result?

Solution

(a) Given a=1-0.95=0.05, Z(0.025) = 1.96 (from standard normal table)

So the lower bound is

p- Z*sqrt(p*(1-p)/n) =0.55-1.96*sqrt(0.55*(1-0.55)/1028) =0.5195879

So the upper bound is

p+ Z*sqrt(p*(1-p)/n) =0.55+1.96*sqrt(0.55*(1-0.55)/1028) =0.5804121

-----------------------------------------------------------------------------------------------------------

(b)Because we use the different margin of error