 So now let's test some actual hypotheses. And so the way we might go about this is that the null hypothesis should allow us to calculate the probability of the observed event, and so we might consider all events of equal or lower probability. For example, suppose an experiment is repeated eight times. If the null hypothesis is the true state of the world, then the probability of observing an outcome K times in those eight trials is given by the following table. You can take these probabilities as being computed in some fashion from the null hypothesis. We won't go into the details of how you'd find these probabilities right here. So we do our experiment, and maybe the outcome occurs six times, and we reject the null hypothesis. Remember, this sets a precedent for when to reject the null hypothesis, and so the question you want to ask is when else should we reject the null hypothesis? So we might proceed as follows. If the null hypothesis is the true state of the world, then the probability of observing the outcome six times is 0.157. So we should also reject the null hypothesis for any observation that has an equal or lesser probability. So if we look at our table, the outcomes with lower probability are these, and so if we observe the outcome seven or eight times, or if we observe the outcome two, one, or zero times. And this leads to the following idea. Suppose we reject the null hypothesis on the basis of the observation. This gives us a set of outcomes, any one of which is grounds for rejecting the null hypothesis. But in a world where the null hypothesis is true, we could also observe one of these outcomes, and this gives us the following idea, what we call the p-value. The p-value is the probability of observing an outcome of equal or lower probability than the observed event under the assumption that the null hypothesis is the true state of the world. So let's go back to our experiment, and suppose we observe the outcome occurring six times, let's find the corresponding p-value. So remember the p-value is the probability of observing an outcome of equal or lower probability than the observed event under the assumption that the null hypothesis is the true state of the world. So if the null hypothesis is true, the probability of observing the outcome occurring six times is 0.157. The p-value will be the sum of all probabilities that are equal or less than 0.157. So those probabilities are these. We'll add them up, and get a p-value of 0.308. Well that's kind of nice, but what does it tell us? Remember our goal is to infer the true state of the world. So our decision could be correct, or it could be wrong. Let's consider our error table. Suppose we reject the null hypothesis on the basis of an observation. If the null hypothesis is true, the p-value tells us the probability we would have seen an observation that would cause us to reject the null hypothesis. Now that's quite a mouthful, so let's think about that a moment. What this is saying is that if the null hypothesis is true, it's possible that we'll see something that will cause us to decide that the null hypothesis is not true. So going back to our idea that a coin is fair, it's possible that a fair coin could land heads eight times out of ten. But based on this observation, we might have rejected the null hypothesis and concluded the coin was not fair. But since the coin was fair, we would have committed a type one error. The possibility that we'll see an observation that causes us to reject the null hypothesis, even when it's the true state of the world, is something we always have to keep in mind. And so what that means is that this p-value corresponds to but is not equal to the probability of making a type one error. And this last is a very important thing to remember, the p-value does not give you the probability of making a type one error. So let's tie everything together, so again we have our experiment, we have our table of probability, and suppose we observe the outcome occurring six times, should we reject the null hypothesis? And so it comes together like this, we found that k equals six occurrences corresponds to a p-value of 0.308, and we might argue as follows, if we reject the null hypothesis, then in a world where the null hypothesis is false, we'd be making the correct decision. But in a world where the null hypothesis is true, we'd reject it incorrectly about 31% of the time. And in this world where the null hypothesis is true, this would correspond to a type one error, the wrong decision. And so here's where we have to make that tough decision, which is ultimately based on the consequences of making an error. But as a general rule, this probability 31% is high enough that we might be inclined to fail to reject the null hypothesis. And we might say that the evidence is not statistically significant.