In this problem, we want to sample 250 voters from across three districts, which are our strata in this example to estimate the mean voter support on an issue. And this voter support is being measured on a scale from 0 to 10. So we want to make sure that we uh collect that. We randomly sample enough people from each of our three strata to make sure that everyone's proportionally represented. And because we have information about the sample standard deviations that were maybe gathered from previous data, uh in those districts, we can use this method of proportional sampling that takes into account the variation and responses. So first thing we need to do is determine the sample sizes for each district. So let's go ahead and calculate this these the number of individuals that I need to randomly sample from district one will be computed by taking n one times S one over and one times S one plus in two times S two plus n three times S three. And then we need to multiply that. Buy em and I realize now that I have an error in the expression that I had up here, this needs to be all times. Mm And when you're doing these problems, if you forget to multiply by m uh, it will be noticeable because you'll get a number that's between zero and one. Um, you should have a whole number that represents a sample size, so we would be able to tell pretty quickly if we had made a mistake. Okay, so N one number of people in district one is 1525. Number of people in the sample standard deviation from that district is 2.2. The sample size m that I want to have total across the district is 250 and then this is going to be divided by 15, 25 times 2.2 plus number of people in district Two times that sample standard deviation So 9, 17 times 1.4 plus 2890 times 3.3. Okay, and what you should get if you multiply that out in your numerator, you'll have see 83 or 838,750. And then in the denominator you should have 14,175 0.8. You might pause the video and do those calculations. This will give you 59 0.1677 If we stop at four decimal places and remember this is a sample size and we need to round to the nearest whole number. So we should sample 59 voters from district one. Okay, let's do the same process for district two. What we're going to have is in two times S two over. And because I've already calculated this denominator and it's not going to change, I'm going to just use the number that I calculated before 1 14,075 10.8. And this is times mm So we should have 917 taking this information from the table times 1.4 times 250 over that denominator. When you calculate this, you should get 22.6407 and rounding to the nearest whole number that will give us 23 voters from district two. And lastly, the number of people we need to sample from district three, that will be N three times S three over that denominator. That we've already calculated times the total sample size we want. So taking this information from the table will have 2890 times 3.3 mm is still 250 keep our denominator the same. And when you calculate that, you should get 1 68.1916 which will round to 168 voters. It's always a good idea to check, let's make sure that we did the calculations correctly and that this does indeed give us the total sample size that we want to have. So we'll add 59 from district one plus 23 from district two from 100 plus 168 from district three. And that will indeed give us 250 voters which is the value of em that we wanted. So everything checks out. You likely did our calculations correctly. So we'll get 59 voters from district one 23 from district two and 168 from district three. Then we're asked to actually estimate the mean voter support using some additional sample data. So what we want to do is estimate the true mean which we represent with the greek letter mu And we have a formula for that. We will need to have some sample mean information. So were given a sample mean from district one of 6.2 sample mean from district two of a 3.1 and a sample mean from district three of 8.5. And the formula that we are then going to use is N one over M times X. Bar one plus end to over M times X bar too plus and three over M times X bar three. Okay, so using our calculations from the first part, we have 59 over 250 times the sample mean given. Which is 6.2 plus 23 over 250 times the sample mean of 3.1 plus 168 over 250 times 8.5. And what you should get when you perform that calculation is approximately 7.4604 So if we wanted to round that to one decimal place, we could say that we're estimating the mean voter response across these districts to be 7.5 on a scale from 0 to 10.