50% of teens believe that we need to address the problem of climate change. If a random poll of 1500 teens was taken (about the size of the Gallup poll), what is the chance that its proportion would accurately reflect the population proportion within 3 percentage points?
Any help is appreciated, I know half of the sample is 750, so am I looking for the chances that within the sample it's between 48.5% and 51.5%?
Note that this assumes you are not using the statistical ideas of normal distribution
What you want is that between 705 and 795. So you want the probability that that X teens choose yes out of 1500. Since there is a 50% chance for each teen, we can use binomial probability. Let's look at a small example: Suppose you flipped a coin 5 times and wanted the probability that it was heads twice. So we know there is a 50% chance of heads, and there are many ways this can work. We can have HHTTT, or HTHTT, just to name a few. In fact, there are 5 choose 2 ways of this working out, which uses some fancy combinations. And there is a .5^5 chance of each of the different ways occurring. So as a final result, you would want $.5^5 * (5 choose 2)$
For your question, you want not just that, but also for 3 Heads.
So you want $\sum_{n=705}^{795} [(1500 $ choose $ n)*.5^{1500}]$. There should exist a function on your calculator that will do this for you. If anyone knows how to LaTeX the (n choose r) formula, that would be helpful to this answer. Hope this helps you.
