Fox News released its final poll for 2016 presidential election: Hillary Clinton

Question

Fox News released its final poll for 2016 presidential election: Hillary Clinton at 48%. Donald Trump at 44%. Gary Johnson at 3% and Jill Stein at 2% based on 1295 persons polled from November 3-6, 2016, with a margin of error of ±2.5 percentage points. The final result showed that Donald Trump won 46.1% popular votes. (a) You see that it is likely, but not certain, that polls like this give results that are correct within their margins of error. So what is the probability that the sample proportion p for an SRS of size n 1295 falls between 0.436 and 0.486? (that is, within ±2.5 percentage points of the true p) (b) What is the probability that the sample proportion p falls between 0.436 anod 0.486 (that is, within 2.5 percentage points of the true p) if the sample is an SRS of size n 500? Of size n 3000? (c) The changing probabilities you found in the previous parts are due to the fact that the standard deviation of the sample proportion p gets smaller as the sample size n increases. If the population proportion is p = 0.461, how large a sample is needed to reduce the standard deviation of p to 0.01?

Explanation / Answer

a)p=0.461 and n=1295

here std error =(p(1-p)/n)1/2 =0.0139

hence probability =P(0.436<X<0.486)=P((0.436-0.461)/0.0139<Z<(0.486-0.461)/0.0139)=P(-1.8048<Z<1.8048)

=0.9645-0.0356 =.9289

b)for n=500

here std error =(p(1-p)/n)1/2 =0.0223

hence probability =P(0.436<X<0.486)=P((0.436-0.461)/0.0223<Z<(0.486-0.461)/0.0223)=P(-1.1215<Z<1.1215)

=0.8690-0.1311 =0.7379

for n=3000

here std error =(p(1-p)/n)1/2 =0.0091

hence probability =P(0.436<X<0.486)=P((0.436-0.461)/0.0091<Z<(0.486-0.461)/0.0091)=P(-2.7470<Z<2.7470)

=0.9970-0.0030=0.9940

c)sample size required n=p(1-p)/(0.01)2 =0.461*(1-0.461)/0.0001=~2485

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