Comparing population proportions 2

Where we left off in the last video, we were trying to figure out if there's a meaningful difference between the proportion of men voting for a candidate and the proportion of women. We sampled 1,000 men, sampled 1,000 women, and we got a sample proportion for each of them. We got 0.642 for the men and 0.591 for the women. But our goal is to get a 95% confidence interval. So just based on our actual sample, we got-- let me write it over here. We got our sample proportion for the men minus-- let me do this in a neutral color. We got our sample proportion for the men minus our sample proportion for the women being 0.642 minus 0.591, that's 0.051. I just subtracted this from that. So what we want to do when we want a confidence interval, we want to be confident. I'll always have to say that because it's not going to be super precise. We want to be confident that there's a 95% chance that this thing right here-- remember, when we took the two sample proportions and took their difference, it's like taking a sample from the sampling distribution of the statistic. So we want a 95% chance that the true mean or the true value of this, that P1 minus P2 is within some range, let's say is within d, I'll say d for distance, of the actual difference that we got with our samples. Within d of 0.051. And I write this multiple times, but I always write it this way. I don't just give the formula that you normally see in books. It's very easy to memorize if you do, but this way, you actually see why this confidence interval makes sense. If there's a 95% chance that P1 minus P2, the actual true proportions, the difference of the true proportions, is within d of the difference between our sample proportions, this statement right here is the same thing that there's a 95% chance that 0.051 is within d of this actual parameter, P1 minus P2, which is the same thing as the mean. So we need to figure out some distance around this mean, where if we take a random sample from this, and this is a random sample from this distribution, it has a 95% chance of being within d of this mean, because if it's within d of the mean, then there's also a 95% chance that the mean is within d of our sample, and then we'll have our confidence interval. Our confidence interval would be this value plus d and this value minus d. So what are these? What is the distance d? Well, in a normalized normal distribution, I got a Z-table right over here, and we can assume everything is normal, especially the sampling distributions because our n is so big and also our proportion is not close to zero or one. It's nice and close to the middle, so we don't end up with all these weird cases near the edges. We say, OK, how do we contain the middle 95%? How many standard deviations in a normal distribution do we need to be away from the mean in order to contain 95% of the probability? Now these Z-tables, and we've done it multiple times, give you cumulative distribution. We're looking for this Z-value right over here. If it's containing 95%, you're going to have 2.5% over here and you're going to have 2.5% over here. So from a Z-table's point of view, this Z-table gives you the cumulative probability up to that Z-value. So what we're looking for is actually 97.5%. We're looking for something that contains all of this over here. If we get the Z-value and then apply it on both sides, then we're going to have something that contains 95%. So let's look up the 97.5. 97.5 is right over there, and that is 1.96 standard deviations. So this is 1.96 for a normalized standard deviation, or a Z-score of 1.96. So if we looked to this normal distribution right over here, this distance that we care about is going to be 1.96 times the standard deviation of this distribution, so it's going to be 1.96 times all of this business. 1.96 times the standard deviation of this distribution. And so we just need to calculate this and multiply it by 1.96. Now, we have a problem. We don't know the true parameters P1 and P2. We don't know the true population parameters. We don't know P1 and P2. That's part of the problem. We're trying to figure out if there's a meaningful difference between P1 and P2. But we've seen it multiple times. Since our sample size is a large, we can estimate P1 and P2 with our sample proportions. So we could change this to approximately and we can use our sample proportions. And we know what those values are. And actually this n over here was 1,000. So let's figure that out. Let's just get the calculator out. It's just going to be one big calculation here. So what we have is the square root, and then in parentheses, our sample proportion for the men is 0.642, and then we're going to multiply that times 1 minus 0.642, close parentheses. That's that over there divided by 1,000. And then we're going to add to that plus-- do the same thing for the women. Our sample proportion is 0.591 times 1 minus 0.591. So that's this term right over here divided by 1,000. Once again, I need to get the parentheses right. And then we just need to close the parentheses, this original parentheses, because we're taking the square root of everything. So we get 0.021, or maybe we'll say 0.022. So this value right here is approximately 0.022. So going back to our question, or this distance that we care about, this value is going to be approximately, or our best estimate of it, is 0.022. So let's just multiply that. 0.022 times 1.96 gives 0.043. I'll just round it. So this right here is equal to 0.043. And just like that, we have our confidence interval. We know that there's a 95% chance that the true difference of the proportions is within 0.043 of the actual difference of our sample proportions that we got. Or if we actually want to get an interval, we take this value minus 0.043. So let's do that. So we could have 0.051 minus 0.043 is going to give us 0.008. And then if we add it, so 0.051 plus 0.043, it gives us 0.094. So the 95% confidence interval between the proportions of men and the proportion of women who are going to vote for the candidate for P1 minus P2 is 0.008 to 0.094. I have it right here on the calculator. And we're done. So it does seem we're confident that there's a 95% chance that men are more likely to vote for the candidate than women.