And so ultimately, what I want to do is just wrap up our discussion of analyzing means when using a confidence interval and a hypothesis test. I want to wrap up this discussion of 9.3 and 9.4 because remember, we studied confidence intervals in Section 9.3 and studied hypothesis testing here in 9.4. I ultimately want to wrap up these two sections by asking the same question again, which is: when will these two statistical inferences give me the same results? When will this hypothesis test give me the same results as a corresponding confidence interval? We answered this question back in Chapter 8, and it turns out the conditions when these two things — confidence intervals and hypothesis tests — will give the same result. These two conditions are exactly the same. Once again, the first condition is that your alternative hypothesis needs to use the "not equal to" sign. The first condition to get matching results between hypothesis testing and confidence intervals is that my alternative hypothesis needs to use the "not equal to" symbol. And the second condition: the sum of the confidence level, like 90%, and the significance level, like 10%, needs to add up to 100%. We saw that before; we saw this back in Chapter 8. We said that when the confidence level and the significance level add up to 100%, we said then that the hypothesis tests and the confidence interval will be complementary. So, ultimately, that is going to be, yet again, the same two conditions we're going to need, even though we're working with means now. If your alternative hypothesis uses a "not equal to" symbol and the sum of your confidence level and significance level equal 100%, we're going to get the same results. We're going to get the same results where, once again, if you are being told to reject your null, it means your status quo will not live in the confidence interval. Once again, we will get that if we're failing to reject the null, it means, again, that the status quo will appear in the confidence interval. Those ideas highlighted in blue and purple were what we identified back in Chapter 8. So, really, the wrap-up here in 9.6 is just kind of a Deja Vu of what we saw in Chapter eight. So, let's see it now apply when looking at means instead. Let's do a practical example of this in Example One. Suppose I ran a hypothesis test and here in the hypothesis test I already gave you guys step one: the null is "Mu is equal to 80", the alternative is "Mu doesn't equal 80", where "Mu" is representing the average height of male college basketball players. And in this alternative hypothesis, I am using a significance level of 5%. Now, I did, in essence, step two for you, in that I told you what the significance level is, so we're good there. And you know what? I also gave you guys step three. I also told you what is the P-value: 0.02. So, ultimately, what I want us to do is ultimately decide: how will this confidence interval, this confidence interval with a confidence level of 95%, how is that going to contain or not contain 80? How is that going to contain or not contain 80, my status quo? So, again, let's just make sure we understand what is going on with this hypothesis test, alright? Ultimately, I want to just really quickly again summarize what were the three steps of this hypothesis test. Again, this is very similar to what we did in 9.4. Step One is done for us in that the null is "Mu equals 80", the alternative is "Mu doesn't equal 80". Step Two, I'm just assuming the conditions are going to hold more just because I want to make sure I get to step four. And again, step three is already done. I already found the P-value, 0.02. So, really, I want to get to the conclusion. I want to get to the conclusion of the hypothesis test where, in this case, when comparing this P-value of 0.02 to the significance level of 5%, "p" is definitely smaller. I reject the null. So, we got to take that null there. We got to take that null expression. H knot is that "Mu equals 80". And that ultimately, to reject this equal sign, is then saying that "Mu doesn't equal 80". To reject the null means that I am rejecting this equal sign. So, rejecting equal means not equal. And notice, not equal is then my alternative hypothesis. So, for starters, why is it that the first condition needs to be a "not equal to" symbol? It's because we need to make sure when rejecting the null, we are going to its exact opposite of not equal to. So, alright, we can see here already I have one of the three conditions to show that they have the same results. I want you to ultimately see here that condition one is already satisfied. Condition one is already satisfied because my alternative hypothesis "Ha" is using a "not equal to" symbol. So, the only other thing we need to check to make sure that these two have the same results is we just need to make sure that the significance level plus the confidence level is equal to 100%. Well, let's see here. My significance level is 5%. My confidence level is 95%. It's going to give me 100% because both conditions are holding. What this is telling me is that this hypothesis test is going to have the same results as the corresponding confidence interval. So, what is that going to mean? Well, let's go back and remember what a confidence interval is ultimately doing for us. And this is what we learned in Section 9.3. Let's go back and remember when it came to looking at this confidence interval. We were creating, quite literally, an interval from a lower and an upper value where the idea of this confidence interval is that any number that falls in this interval is representing a plausible value for "Mu". Let's remember that a confidence interval is representing possible values of "Mu", just in general. That is what a confidence interval represents. It represents what could possibly be my population mean. And so, what does this middle red portion give me? What this middle red portion gives me is that the results of this alternative hypothesis then is going to apply into this confidence interval. See what these two conditions are implying is that this results from my alternative hypothesis, "Mu cannot equal 80", then absolutely needs to apply in my confidence interval. So, therefore, I'm quite literally going to copy and paste this over here with my confidence interval. So, ultimately, because we showed that the hypothesis test will have the same result as the confidence interval, what this is telling me is that in this confidence interval, we are then going to have this result where "Mu cannot equal 80". What is that telling me? It's telling me then 80 is not going to be in this confidence interval. That is, 80 cannot be in the confidence interval. And so, that is then the power of what we are seeing in this wrap-up. In what we are seeing in red is that you can run one test like a hypothesis test, which is frankly a little easier to run. We all probably feel it that hypotheses tests are easier. And that if these two conditions are met, you then can apply the results of that hypothesis test into the confidence interval. And notice, that's exactly what we saw here. That if you reject the null, which is what we did, if you reject the null, what that means is that status quo value, which was 80, does not appear in the confidence interval. And so, I want you guys to see that this is the final wrap-up of one population and when we're studying confidence intervals and hypothesis testing, that those two will ultimately merge together when these conditions are satisfied: the alternative uses "not equal", the sum of the significance level and confidence level are 100%.