00:01
If we let n be the number in our sample, number in sample, then we take a sample of the population, and we let x be the number who support the candidate, and we let p be the proportion of people in the population, proportion of people who support the candidate, then your x is distributed binomial with size n and proportion or probability of success p.
00:52
So then p hat, which if we let that be x over n, is distributed normal with mean p, which will be expected value of x over n, but that's p.
01:10
And your variance or standard deviation here for p hat will be the square root of p times 1 minus p over n.
01:22
So if we're looking for a 95 percent confidence interval for the true population proportion, that will be p hat plus or minus the z 0 .025, because alpha here is 5 percent, so alpha over 2 is 0 .025, and we multiply that by p times 1 minus p over n.
01:54
Well, this will always be less than or equal to p hat plus or minus z alpha over 2, when alpha over 2 is that, it's going to be 1 .960.
02:12
So this will always be less than or equal to 1 .960 times the square root of 1 fourth over n, since p times 1 minus p is less than or equal to 1 fourth for all p in the interval from 0 to 1.
02:38
So that means that our margin of error will be less than or equal to 1 .960 times the square root of 1 over 4n, and the square root of 1 over 4n is 1 .960.
03:03
We can split that up as square root of 1 over n and pull out the square root of 4, which gives us 2...