Once again, welcome to a new problem. This time we're dealing with proportions. And when you think about proportions, you could have a population and the population would have a population proportion population proportion and the population proportion would be something like the proportion of females in college campus, the proportion of females in a college campus and the population proportion is based on capital P. Um as opposed to the sample proportion, which is jihad, so p heart is X over and this is the sample proportion where you take the uh number of subjects of interest divided by the sample size. And for the most part, there's always a likelihood that you want to do an estimation of the population proportion based on the simple proportion and the estimation can have a margin of error and a confidence level. So for example, if you're 95% confidence Uh then it means your error is 5%. Or sometimes we call that alpha is 5%. And from the distribution point of view, what you're saying is that this middle part is .95 improbability. And then you split the tails, you get .025 on the left tail and then .025 on the right tail. And so the formula for confidence interval uh is the same as the hot plus minus margin of error. Where the margin of error is the same as the alpha over to Radical P Hut one -P had all of the end Members the alpha of the two For 95% confidence. From the tables would be 1.96. So that's the Z score. These twosies course, this would be the negative Z score and then this would be the positive Z score. That's why we have plus and minus over there. Uh So coming back to the problem, coming back to the problem, we have a situation where we're thinking about the report. So given, given the 2003 statistical abstract of the U. S. We see that uh the percentage of mm hmm smokers above 18 and what we're doing and this statistical abstract is too Uh huh. A design study I designed study collecting data own smokers and non smokers on smokers and nonsmokers. Mhm. And there's an estimate, there's an estimate of the proportion who smoke uh early there's an estimate of studies. So in part a. Mhm. Are determined the sample size becoming the sample size to estimate the proportion of smokers and the population uh with 95% confidence level And uh .02 error margin of error. So that's what a And then B mm hmm. Using using sample size requirements, whole suggestions if you want to call it that from part a mm. The study finds 5 20 smokers. Mhm. What is the points estimate of proportion of smokers? Mhm. In the population and the population. And then but see uh determined The 95 confidence interval for smokers in the population, 95% confidence interval of smokers uh in the population. So I want to jump right into it. We have the numbers to work with this problem. So we'll start by saying, Well this was actually 30%. So just remember that Start by saying the P hard in part AP heart is 30%, which is the same as .30. The margin of error is point to remember if you have the hard plus minus margin of error at the same 0.0.30 plus minus point or two. Uh and then your confidence level is 95%. And that means that if you're looking at a distribution, this middle part is .95 And then we split the two sides. This other side is .25 and this other side is also .25. And so he is the alpha over two is the same as 1.96. So we go back and say the margin of error is the alpha over too radical P hut one minus p hurt Oliver in, I want to solve for end. So we divide both sides by Uh zero off over to these two cancel out and we can switch it. So we have radical P heart one minus P hard all over in this is E of zero off over to the square both sides of the equation. And we end up having p hard one minus P hut Oliver in um This is going to give us E squared of a Z off over two squad. And then we multiply both sides by N. And also multiply both sides by the reciprocal of E squared ZR for over two on both sides. That way we can cancel out the uh we can cancel out the sample size is on this side. Then you can cancel out these two on this side. So now we have an appropriate formula for the sample size. So we have ZF over to uh, squirt and then we have a jihad one minus P hurt all of E squared. So if you go back to part A, you're going to see uh want to determine the sample size to estimate the proportion of the population, given that confidence level. So given that confidence level. So, uh, assuming, assuming p heart is 0.45. So we're doing the estimates and assuming Prd's uh huh 0.45. What's going to happen is that I'm going to get And being the same as 0/2 of the two squared and this part Becomes .025. And the reason why it's .25 is because if this is .5 And this fight is also .5, You get the product and you get .025. And then you want to divide that by obviously E squared. The next step would be to plug in the specific numbers 1.96 sq Times .025. And then we want to divide that by e squared, which is a point or two squared. And it's going to give us a sample size of approximately uh 2000 and 17. So that's the first part of the problem. In part a uh in the second part will give an X to be 520 and remember the sample size is to 17. So if you go back to the second question is using that sample size that we just got approximately 2017. The study finds 520 smokers. So what's the point estimate of the proportion? So what's the pr come back here? We know that he had is the same as uh X over. N Soapy hot is going to be the same as X being 5 20 And being 2017. So we get the p had to be equivalent to mm hmm, Approximately 0.25 78. Then the last part of the problem, this is Patsy. Uh we want to find the confidence interval and the confidence interval is p hard plus minus margin of error. already have the P had .2578 and then plus minus the margin of error which was mhm zero for two. So the margin of error. Was he off over to radical uh points. We actually want to displace it here. Zf over to body called p hut one minus p hut all over in. So we're gonna place it here Z alpha is 1.96 and then Radical P Hot is the same as 0.2578 And then 1 -1 0.25 78 11 is still going to 578. And then we want to divide that by 2017 and you can see that the confidence Interval and this is at 95%. If you simplify that On the lower side, it's going to give you 0.2387 and then on the Apple side is going to give you 0.2769. Hope you enjoyed the problem. Feel free to send any questions or comments and have a wonderful thing.