 Welcome to our lecture on understanding hypothesis testing. We're going to try to give you a deeper understanding of stating the null and alternate hypotheses, substantiating a claim, as well as what we're actually doing when we use sample evidence to test our hypotheses. This is an optional topic for those of you that want to understand, have a deeper understanding of what's going on when you do a hypothesis test. In the past, when we tested hypotheses, we took a very regimented, very prescribed step-by-step approach. You might even call it a cookbook approach. We knew what we were doing, but we didn't really have a deeper understanding of what we were doing. We set up a hypothesis, we set up a decision rule, we looked at the empirical data, and then we compared the empirical data to the decision rule and came to a conclusion to either reject the hypothesis or not. In this lecture, like I said, it's an optional lecture, in this lecture what we do is we try to explain in a little bit more depth what's going on under the hood, what we're really doing when we do hypothesis testing. Well, here's an interesting question. Is it easier to prove that something is false or something is true? And of course, it's easier to prove that something is false than true. In fact, the example that we give is you make a statement that all geese are white. All I need to refute you is to find one goose that's not white, one black goose and you refute it. On the other hand, to prove that it's true, to prove that something is true, especially a statement like that that all geese are white, I have to find millions and millions of geese and you still refute me. I can show you 27 million geese that are white and you produce the one black goose and you've cooked my goose, so to speak. Well, we have the same issue here when it comes to setting up HO and H1. It's different depending on who is doing the testing. If you want to substantiate your claim and show that you have a product that has a certain life, that's called substantiating your claim, that's going to be a little more complicated in terms of HO and H1 than if somebody's trying to refute this. So we have two ways. We're going to set up HO and H1 differently. If we're trying to substantiate a claim, suppose my claim is that my computers will last at least 10 years. That's substantiating my own claim. Refuting a claim, on the other hand, is somebody, usually the competition or government, somebody's trying to show that I've lied and they want to refute what I've said. So we have a difference you'll see in a moment between substantiation, which is called confirmation of a claim, or refuting the claim, refutation. We're going to see in a moment that when it comes to substantiating a claim, you want to set up your HO and H1 in a way you can be extremely conservative. You want to make sure that there's a great deal of evidence that your claim is accurate because if you're mistaken, then you lose your reputation, you get a lot of trouble. So you're going to set up HO and H1 in a very conservative way, and actually you're going to see that your claim is going to be H1. Whenever a company wants to substantiate a claim about its product, it's going to make sure that the claim is in the H1. In this example, we're going to look at a company that makes cables. And these cables have to have a breaking strength of at least 200 pounds. So we want to substantiate the claim because obviously our customers will be very upset if we promise them that these cables will not break unless you use much more than 200 pounds of pressure. We want to make sure we're right about that claim. These cables have that kind of strength. So we're going to see how we're going to set this up with HO and H1. As I noted, when you substantiate a claim, the company's trying to substantiate its claim, that becomes the H1. So notice H1 is that mu is greater than 200 pounds. That's what the company's claiming, that the breaking strength of its cables is 200 pounds or more. So notice that's the H1, which of course means that HO now is that mu is less than 200 pounds. Okay, so we're going to use an alpha O5. And on the previous problem, this problem, we looked at the sample evidence and notice we get a Z of 3.33. We reject HO. Rejecting HO here is good because we reject the HO that it's less than 200 pounds breaking strength. And we set it up purposely this way. We want to reject HO being very conservative. We want to reject HO and then be left with H1, which is that that's our claim. Again, when you substantiate a claim, H1 becomes the claim. Here you can read exactly what I said that when you substantiate a claim, that becomes H1. So again, when a company is trying to substantiate a claim, it makes H1 their claim. Now, when somebody wants to refute a claim made by a company, let's say it's the competition or government. They want to refute the claim. They're going to put the claim in the HO. HO becomes the claim. They're going to collect the sample evidence trying to refute your claim. But HO is the company's claim and the competition is hoping to refute it. Okay, the competition is trying to refute the claim of this cable company. I know the company claimed that their chains, cables have a breaking strength of 200 pounds. So that's HO. Mu is greater than 200 pounds. That's the claim. Notice it's in the HO. And now they're going to take sample evidence. They're going to test at the 05 level. Notice the critical value for Z is minus 1.645. And they're going to take the sample evidence. Now, again, if they get a sample mean of 220 pounds, they don't have to do anything. They know right away they don't have the evidence. They only can reject HO if they have a number substantially below 200 pounds in terms of the sample mean. But if the sample mean is 220, that supports the claim. They know they have no way to reject HO. We're going to try to give you a deeper understanding of hypothesis testing, looking at it in a slightly different way. This is what we've been doing. The company makes a claim that its tablets have a life of at least eight years. Okay, you want to test the claim. All right. So what do you do? You took a sample of 100 tablets made by this company. Now you find that X bar, the sample mean is 7.6 years with a standard deviation of 1.2 years. Now, before you reject this claim, you want to make sure you're not looking at sampling error. 7.6 is less than 8, but it may not be significant. It could be sampling error. So we set up HO. HO is the claim. The company claimed eight years or more. The mu is greater than 8. H1 is mu is less than 8 years. We take the sample evidence, convert it into a Z score. We ended up with minus 3.33. Our critical value was minus 1.645. And we reject HO at the probably less than 05 level. This is the way you've been taught to do it. It's good. It works. Your conclusion is reject HO, and you're not going to buy these tablets from this company. You're going to say there's no way these tablets have an average life of eight years given your sample evidence. We're going to explain it a little differently now. When a company makes a claim about a parameter, in this case, that the lifetime of its tablets is eight years or more. That's mu is the parameter. So first of all, we start with temporarily accepting the claim. We call that a straw man, something we want to shoot down. But right now we accept it temporarily. We say mu is more than eight years. Now, how do we shoot it down? We're going to look at the sample evidence and see if the sample evidence supports this claim. In a sense, what we're doing is we draw the distribution, the Z distribution, with our straw man there. Notice the center, the mean, median mode, the center of the distribution is eight. Because that's the straw man. We said, okay, you claimed it's eight years or more. So we're using your eight. And now we have a rejection region. There's only a 5% chance of happening. You say, well, where's our sample evidence going to bring us? Will it take us into the rejection region? In which case we're going to reject everything, your HO, the straw man. But for the moment, we've accepted your claim. That's why it's there, that eight. If that distribution that you're looking at, with eight at the center, that's the straw man. We've accepted it temporarily. Our sample evidence results in a Z value of minus 3.33. Using the cumulative Z table, we see how much area is between negative infinity and minus 3.33. Note that it's 0.0004. In other words, there are only four chances in 10,000. So what does that four chances in 10,000 represent? That's the likelihood of getting the sample evidence of 7.6 years, or something even less, like seven years, because we're going all the way to infinity. Anything less than 7.6 years. So we had the straw man set up. We say, well, is this a sample evidence one should see? Well, if the population mean is eight years, the likelihood of getting sample evidence that we got, 7.6 years, or even worse than that, turns out to be only four chances in 10,000. Right? 0.0004. That's a lot less than 5%. In other words, this is not the sample evidence one should be getting if your claim is true. Remember that was a straw man, eight. We drew the distribution. Well, the sample evidence doesn't support this, and we're going to shoot down the straw man. Okay, let's try the same problem where the sample evidence leads to a different conclusion. Again, the company's claim is the same. Their tablets have a life of, again, we spoke about the parameter of at least eight years, and H1 is that it's less than eight years. Now, look at the sample evidence, 7.9 years. We took a sample of 100 tablets randomly, and we found that X bar was 7.9 years. The standard deviation is 1.2. Now our Z value is negative 0.83. Normally, we say don't reject. We're not in the rejection region. What does that mean? In the cumulative Z table, you'll find the area between negative infinity and minus 0.83. This is the cumulative Z table as an area of 0.2033. It's telling us that the likelihood of getting that kind of sample evidence is 20 chances in 100. In other words, it's not so unlikely to get the real mean could be eight of straw man. The real mean could be eight, and getting 7.9 is not so unusual. That's why we don't have the right to reject HO. In other words, we can't say the sample evidence is so unlikely. It could very well happen that the population mean is eight, and you take a sample of 100 and you get 7.9. So essentially, we don't have the evidence to reject HO. So note that the HO is our straw man. So we stop hoping we can knock it down looking at the sample evidence, but the sample evidence didn't give us the evidence that we needed to knock down the straw man. So we have no evidence that the company's claim is false. This is why we don't reject. Technically, we've been using words accept, but really it's not reject. Again, if you do this by computer, the computer just gives you the p-value. It's giving you the likelihood. Again, it's using a cumulative distribution, and it's giving you the likelihood of getting the sample evidence or something more extreme, and you just compare that with alpha. So if alpha is, let's say, 05, and your p-value is 30%, you can't reject, telling you it's a 30% chance of getting the sample evidence or something more extreme. If alpha is 05 and you get something way below 05, your p-value, let's say, is 00001, you will reject. And when you work with a computer, you just need to look at the p-value. Thank you for attending our lecture. As always, we advise you to do lots and lots of problems in order to make sure that it becomes part of you, and you remember your work for now and for the future.