Course Notes - Department of Mathematics and Statistics

Fred for MayorFred has decided to throw his hand in the ring for mayor in the upcomingelection. Fred thinks he needs about 45% of the vote to win. Aftertalking with his 10 friends he found that 8 of them would vote for him,how confident should Fred be based on this polling?This question requires us to build a confidence interval for a proportion,the formula for this isp ± 1.96 ×So if we substitute our values in, we get0.8 ± 1.96 ×Which gives us the confidence interval√p(1−p)n√0.8(0.2)10(0.5521, 1.0479)We can see that this confidence interval is entirely above 0.45 thereforethere is evidence that Fred will get the 45% he requires for victory. Wecan also see though that the width of the confidence interval is 0.4958,so almost 50% which is very large indicating that the sample size is toosmall.Fred thought that perhaps he should poll some more people since hisfriends might be biased, but polling is expensive, he wanted to knowwhat is the smallest number of people he can poll to be sure of theresult within 3%.This problem causes a lot of issues for people, but with practise itbecomes less daunting. The first thing to do is to look at the formulafor the 95% confidence interval.p ± 1.96 ×√p(1−p)nIt is clear the actual width of the confidence interval is dictated bythe expression on the right side of the ±, so this is what we need tomanipulate in order to find the desired sample size.115