So far, we've used two inferential techniques, hypothesis tests and confidence intervals. And it makes sense that the results of these two techniques agree with each other. And they will, if we're using equivalent levels of significance and confidence. So in this video, we're going to discuss the interplay between significance levels used in hypothesis testing as well as confidence levels used in construction of confidence intervals. Broadly we can say that a significance level and a comp confidence level are complements of each other. Think about the most commonly used significance level, 5%, and think about the most commonly used confidence level, 95%. So there, it is not a coincidence that the sum of those two numbers adds up to one. They are indeed complements of each other. However, whether this compliment rule works or not depends on whether we're doing a one-sided hypothesis test or two-sided hypothesis test. So let's look over here at a two-sided hypothesis test where we have an alpha of 0.05. If you have an alpha of 0.05, that means that at each tail you can afford to have about 0.025 of a tail area, so that the total of those tail areas add up to 5%. And usually, when we're thinking about confidence intervals, we're always thinking about the middle whatever percent of the distribution. So if you're thinking about a 95% confidence level, we're interested in the middle 95% of the normal curve. Therefore, a two-sided hypothesis test, with alpha equals 0.05, where the two tail areas add up to 0.05, is indeed going to be equivalent to a 95% confidence interval. If we add up the 95% in the middle with the alpha on the two tails, we get to the one, which is the the entire area under the curve. So indeed, for two-sided hypothesis tests, the significance level and the confidence level are complements of each other. What happens when we have a one-sided hypothesis test though? In this case, we're looking at a one-sided hypothesis test with alpha equals 0.05. I've chosen to include the the tail area on the higher end. We could have done it on the lower end as well. That doesn't matter. But the important thing is that it's either on one end or the other end, right? So, we're looking for where our p value could be anything up to 5% and it would only be in one tail and we could still reject to, the null hypothesis under this scenario. While the lower tail in this figure is of no interest to us within the framework of the hypothesis test. Since confidences always have to be symmetric, right? And the confidence level is always about the middle whatever percent of the distribution. And we cant have a confidence interval that only goes a certain amount of distance in one direction but more in the other direction, we are actually going to need to think about the 5% in the lower tail, even though we are not going to use it for any sort of decision making within the hypothesis test. So if you have a one-sided hypothesis test with an alpha equal to 0.05, your equivalent confidence level is actually going to be 90%. Because we're allowing for the 5% at the one end that you're interested in, and we're having to take care of the other 5% at the other end that you're not interested in, but we need to take into account so that the confidence interval can be symmetric. So in this, case with a one-sided hypothesis test, we can't anymore really say that the significance level that we're using and the confidence level that we're using are complements of each other directly. We can still use that idea though to figure out how to get from one to the other, and really, once again, the key is to always draw your curve and once you draw your curve and mark what you're interested in, that's going to kind of allow you to think about do I need to think about the other tail or not? And if you're doing a one-sided hypothesis test, you will need to think about the other tail, and then you can easily arrive at the confidence level that you need. Why is this of importance? Oftentimes, you may want to use both methods when you're doing inference, and it doesn't make sense for your methods to not agree with each other. So, for the most part, if you have the given significance level and you have accurately determined the equivalent confidence level, the results from the two approaches should always equal agree with each other. So, to summarize what we've gone through, a two-sided hypothesis with a threshold of alpha is equal to a confidence interval with 1 minus alpha. So, in this case, if your hypothesis test is two-sided, your confidence level and your significance level are compliments. A one-sided hypothesis, with a threshold of alpha, is equivalent to a confidence interval with 1 minus 2 times alpha. If the null hypothesis is rejected, a confidence interval that agrees with the result of the hypothesis test should not include the null value. This, hopefully, makes sense. Because if you're saying that you're rejecting a null hypothesis, the null value, then, should simply not be in the confidence interval. Otherwise, you would be contradicting yourself, saying that it's a plausible value for the parameter of interest. Similarly. If the null hypothesis is failed to be rejected, a confidence interval that agrees with the result of the hypothesis test should, indeed, include the null value. So using what we've learned about the equivalency of the confidence and significance levels, you can determine which level to do which technique in. And then, using the two bullet points at the bottom of this slide, you can determine whether the results of the two techniques agree with each other. Moral of the story is, if your confidence interval includes a null value, don't reject it. If your confidence interval does not include the null value, then you can go ahead and reject it.