Hypothesis test comparing population proportions

In the last couple of videos we were trying to figure out whether there was a meaningful difference between the proportion of men likely to vote for a candidate and the proportion of women. And in the last video, we actually estimated that using a 95% confidence interval for the difference in the proportion of men and the difference in the proportion of women. What I want to do in this video is just to ask the question more directly. Or just do a straight up hypothesis test to see is there a difference? So we're going to make our null hypothesis. No difference. No difference between how the men and the women will vote. Or another way of viewing, it that the proportion of men who will vote for the candidate is going to be the same as the proportion of women who are going to vote for the candidate. Or another way you could say that, is that the difference P1 minus P2, the true proportion of men voting for the candidate minus the true population proportion of women voting for the candidate is going to be 0. That's are our null hypothesis. Our alternative hypothesis is that there is a difference. Or that P1 does not equal P2. Or that P1 minus P2, the proportion of men voting minus the proportion of women voting, the true population proportions, do not equal 0. And we're going to do the hypothesis test with a significance level of 5%. And all that means, and we've done this multiple times, is we're going to assume the null hypothesis. And then assuming the null hypothesis is true, we're going to figure out the probability of getting the actual difference of our sample proportions. So we're going to figure out the probability of actually getting our actual difference between our male sample proportion and our female sample proportion. Given the assumption that our null hypothesis is correct. And if this probability is less than 5%, if this probability is less than our significance level. So if the odds of getting these two samples and the difference between those two samples, is less than 5% percent, then we're going to reject the null hypothesis. So how are we going to do this? So if we assume the null hypothesis, what does the sampling distribution of this statistic start to look like? Well, if we assume that the true population proportions are actually the same between men and women. If P1 and P2 are actually the same, then this right here is going to be 0. So what we can do is, we can figure out that we got when we took the proportion of men and we subtracted from that the proportion of women-- So this is our sample proportion of men who, at least in our poll, said they would vote for the candidate. This is a proportion of women who said they would vote for the candidate. The difference between the two was 0.051. So we can do is figure out what's the probability? Assuming that the true proportions are equal, that the mean of the sampling distribution of this statistic is actually 0, what's the probability that we get a difference of 0.051? So what's the likelihood that we get something that extreme? And what we're going to do here is just figure out a Z-score for this. Essentially figure out how many standard deviations away from the mean this is. That would be our Z-score. And then figure out, is the likelihood of getting a standard deviation, or that extreme of a result, or that many standard deviations away from the mean, is that likelihood more or less than 5%? If it is less than 5%, we're going to reject the null hypothesis. So let's first of all figure out our Z-score. So we're assuming the null hypothesis, P1 is equal to P2. Our Z-score, the number of standard deviations that our actual result is away from the mean, the actual difference that we sampled in the last few videos between the men and the women was 0.051. And from that we're going to subtract the assumed that mean. Remember, we're assuming that these two things are equal. So the mean of this sampling distribution right here is 0. So we're just going to subtract 0. And then we have to divide this by the standard deviation of the sampling distribution of the statistic right here. P1 minus P2. Now, what's the standard deviation of the distribution going to be? In the last video, we figured out that we could represent it by this formula over here. But with our null hypothesis, we're assuming that P1 and P2 are the same value. Let me rewrite it. So in our last video, and I don't want to confuse the issue, because in the last video, I made this approximation over here. So let me write the clean version down here. We know that the standard deviation of our sampling distribution of this statistic of the sample mean of P1 minus the sample proportion, or sample mean of P2, is equal to the square root of P1 times 1 minus P1 over 1,000, plus P2 times 1 minus P2 over 1,000. We've seen this in several videos. But in the null hypothesis, we are assuming that P1 is equal to P2. That's what we do. We assume the null hypothesis and see the probability of this occuring. So if P1 is equal to P2, we can just represent them as just some true population proportion. So we could write it like this, the square root of-- we can literally just factor out 1/1,000 times P times 1 minus P, plus P times 1 minus P. Because they're going to be the same value. That's what we're assuming in the null hypothesis. And so this is just two of these over here. So this is going to be equal to 2P times 1 minus P, all of that over 1,000. And we're going to take the square root of that. Now this is the standard deviation, once again, of the distribution of this statistic right over. The sample proportion for the men minus the sample proportion of the women. Now, we still don't know this. We still don't know the true proportion. But we can estimate it using our samples. And since we're assuming that the men and women, that there's no difference between them, we can actually view it as a sample size of 2,000 to figure out that true proportion. So we can actually substitute this with a sample proportion. And we can pretend like our survey of the men and women is just one huge survey. So you have your sample proportion, we're surveying a total of 2,000 people. 1,000 men and 1,000 women. But we're assuming that they're no different. That's what our null hypothesis is all about, assuming there's no difference between men and women. And we got 642 yeses amongst the men and 591 amongst the women. So we got a total of 642 plus 591. If you viewed it as just one huge sample of 2,000 people, we got 642 plus 591 is equal to 1,233 divided by 2,000 gives us 0.6165. And this is our best estimate of this consistent population proportion that is true of both men and women. Because we are assuming that they are no different. So we can substitute this value in for P to estimate the standard deviation of the sampling distribution of this statistic right over here. Assuming that the proportion of men and women are the same. Or the proportion that will vote for the candidate. So let's do that. It's going to be the square root of 2 times P, which is 0.6165, times 1 minus P, 1 minus 0.6165, divided by 1,000. Let make sure I got it. 2 times 0.6165, that's that P right there. Times 1 minus P divided by 1,000. We're taking the square root of the whole thing. And so we get a standard deviation of 0.0217. Let me write this over here. So this thing right over here is 0.0217. So if we want to figure out our Z-score, if we want to figure out how many standard deviations the actual sample that we got of this statistic right over here. If we want to figure out how many standard deviations that is away from our assumed mean, that there's no difference, then we just divide 0.051 by this standard deviation right over here. So let's do that. So we have 0.051 divided by this standard deviation, and that was our answer up here. So I'll just do divided by our answer. And we are 2.35 standard deviations away. So our Z-score is equal to 2.35. So just to review what we're doing, we're assuming the null hypothesis, there's no difference. If we assume there's no difference, then the sampling distribution of this statistic right here is going to have a mean of 0. And the result that we actually got for the statistic has a Z-score of 2.34. Or this is equivalent to being 2.34 standard deviations away from this mean of 0. So, in order to reject the null hypothesis, that has to be less probable than our significance level. And to see that, let's see what the minimum Z-score we need to reject our hypothesis. So let's think about that a second. I'll go back to my Z-table. We want to have a significance level of 5%. Which means the entire area of our rejection, in which we would reject the null hypothesis is 5%. This is a two tail test. An extreme event on either far above the mean or far below the mean will allow us to reject the hypothesis. So we care about the area over here. And over here we would put 2.5% and over here we would have 2.5%. And we would have 95% in the middle. So we need to find this critical Z-score, critical Z-value. And if our Z-value is greater than the positive version of this critical Z-value, then the odds of getting something so extreme is less than 5%, assuming the null hypothesis is correct. So then we can reject the null hypothesis. So let's see what this critical Z-value is. So essentially we want a Z-value where the entire percentage below it is going to be 97.5%. Because then you're going to have 2.5% over here. And we've actually already figured that out. This whole cumulative has to be 97.5%, we did that in the last video. If you look for that, you get 0.975 right there. It's a Z-score of 1.96. I even wrote it over there. So this critical Z-value is 1.96. So what that tells you is there is a 5% chance of sampling a Z-statistic greater than 1.96, assuming the null hypothesis is correct. Now, we just sampled a Z-statistic of 2.34 assuming the null hypothesis correct. So the probability of sampling this, given the null hypothesis is correct, is going to be less than 5%. It is more extreme than this critical view Z-value. It's going to be out here some place. And because of that, we can reject the null hypothesis. I'm sorry for jumping around so much in this video. I had already written a lot. So I just kind of leveraged what I had already written. But since the odds of getting that, assuming the null hypothesis, are less than 5%, and that was our significance level, we can reject the null hypothesis and say that there is a difference. We don't know 100% sure that there is. But statistically, we are in favor of the idea that there is a difference between the proportion of men and the proportion of women who are going to vote for the candidate.