 when you see statistical testings where 95% confident or there's a 5% margin of error in these kind of terms, those are gonna be more technical statistical terms. We'll talk more about in future presentations. Right now we wanna get down the idea of taking the sample data and using that to infer onto the entire population. Once we get good at doing that, once we have that conceptually down, what we would like to do is to get as clear in terms of mathematical tools as possible using statistics so that in probability in essence, so that we can be more confident, right? Then if we can use statistical equations, then we usually have more predictive power and confidence going into the future. So probability theory is a theory in statistical inference. So probability theory is the backbone of statistical inference. It provides the language and the mathematic tools needed to quantify uncertainty and to make educated guesses. So we're using the mathematical tool when we're looking at statistical inference of probability theory. So it provides a means to quantify how likely or unlikely the observed data would be assuming a particular statistical model is true. So note what we have to do if you look at something from a scientific kind of approach, usually if you go into a laboratory, for example, what they're trying to do is isolate everything that has an impact to a few different impacts on a particular item, whatever they're testing, right? So that then they can see the cause and effect of the one thing that they're looking at. So they're trying to isolate everything so that they can look at the cause and effect. When we look at kind of predictability in the real world, we have to make similar kind of assumptions. We have to basically say, well, here's the statistical model that I'm putting together. Certain assumptions are gonna be made in the statistical model to make projections, predictions about future outcomes, for example, in an election. And if these assumptions are true based on then, then we can come up with a mathematical approach of what the result will be. But of course, a model is just a model. So a model is not the real world, because usually in the real world, unlike with a lab, we can't trim everything down to just a couple factors, to test a couple factors. So we have to make, of course, assumptions. And then the question is, is the model that we put together, does it have good predictive power for the results for the entire population or not? So it's gonna be dependent on the model and no model is perfect because the model is not the real world. It's just the model. Hypothesis testing. Hypothesis testing is another key element of statistical inference, where in essence, we form two opposing hypotheses about the population, the null hypothesis and the alternative hypotheses within collect data and compute a test statistic. So clearly, hypothesis testing is a core scientific tool. If we were to go back into our laboratory, for example, and run a scientific test to see whether or not a particular element causes a fluid to turn green, what we would want to do is try to remove all other factors on the fluid, form a hypothesis that being the null hypothesis, nothing's gonna happen when we add the elements to the fluid and then the alternative being that it changes colors, that it turns green, for example. And then we're gonna run multiple tests from that point to see what happens is the general idea. So depending on the value of this test statistic, we decide whether to reject the null hypothesis in favor of the alternative. This process allows us to make statistically informed decisions about the population based on our sample data. So let's take a look at an example here. So we want to be determining the fairness of a coin. That is whether it is equally probable for the coin to land on heads as it is to land on tails. So in other words, we're given a coin a quarter, for example. And if we flip the coin multiple times, we would expect that there's gonna be an equal number of times that it's gonna have heads versus tails or at least the probability of it landing heads and tails should be equal. That would be the assumption we would have and that would be basically our null type of assumption. And the null assumption oftentimes would be things are gonna be as is, right? The standard type of assumption. And then it would be more unusual generally if the coin did not land statistically speaking on a 50-50 chance. So what are the chances that the coin is not fair? That it's more likely to land on heads, for example, than on tails. Well, what would we do in order to test this in the real world? We can't flip the coin infinitely many times because so we don't know all the entire population of the coin flips.