[MUSIC] So here we're going to talk about the limitation of point estimation. So we've already seen if we've taken the population that big and only sample a piece of this population, we will be able to find, let's say, an unbiased estimator, as we've already seen, of the population mean. So in one sense, it's good that if we do the procedure correctly, the estimator will be unbiased. But what do we have? We have a point estimate that's saying this based on this sample that we took, this is our mean. But how reliable is this mean? This is a very big question that we need to ask ourselves. We know that there are some limitations of this mean that we take in our sample. So if we say, for instance, we take a survey or we're talking about microeconomic activities, such as we would like to estimate inflation rate, unemployment rate, or if we're talking about pharmaceutical rate of a success in treatment, all these are just estimators. And there might be even unbiased estimators. But what is the real value? Can we somehow say what will be this value? Because we can say that on average, this is indeed the case. But we know that we will not be able to sample every single piece of this population. We cannot do that. Limitation of resources, limitation of time, and many other constraints that prevent us from doing so. So we have to work out somehow with this value. The confidence interval will be able to assist us in saying something about how confident we are in this estimator. And we need to think this very carefully about this estimator. We know first of all that the population mean is a very reliable estimator. So we know that this is an unbiased estimator, first of all. But we know that it is very unlikely that it will hit the actual mean. We can say about something about the mean, though. It's not that we cannot say anything about the mean, but we can say something about the mean. So the question is, how confident we are that the mean will be somewhere near this estimator? And this is where the confidence interval will play a significant role. So first of all, we have to determine with what error size or what percentage of error that we are happy with, whether it will be 5%, whether it will be 1%, or be it 0%. It will determine the level of confidence or confidence interval that we will have to take into account. You have to be very careful in how you interpret the confidence interval. So the confidence interval is not saying that I'm, say, 95% confident that the mean will be in the specific interval of the confidence. The correct interpretation will be that the confidence interval that you build has 95% probability that it contains the true value of the mean. And this is what you have to work in many ways. So in order to give some content to the confidence interval, let's have an example. In this example, there's a manufacturer of breakfast cereal that is interested in knowing whether the population mean weight of a box cereal is actually 300 gram as specified on the box. We're all familiar with this problem that we see a product that it says that the product weight or the product content is a specific content. But how specific it is, how correct it is, this is the question. The company will decide this based on a sample of at least 30 observations that have a sample mean value of 301.5 gram. Can we use this observation in order answer manufacturer's question? So in this case, what we see first of all is that the mean is 301, and this is the point estimation that we have in the sample. What we need to know is that the mean, the actual mean of 300 that is specified on the box is the correct one based on those 30 observations that are taken. The first question you need to ask yourself is, how confident you want to be in this estimation, or in other words, what will be the error? In order to answer this question, you need to specify what is the level of error that you are comfortable with, whether it's 5%, whether it's 10%, whether it's 1%. And it will affect the level of confidence. So in this example, we've seen that the average value that was given by sampling 30 boxes was 301.5 grams. The next thing we have to determine ourselves is what's the confidence that we want to be in, whether it's 90% confidence, whether it's 95% confidence. And this will determine the level of the confidence interval that we're talking about, where mu, the actual unknown mean, will lie in. So just to give you an example, mu will be somewhere around the x, the mean of the sample, plus minus some value. So this is the value that we will be interested in right now. And this value will be determined by how confident we want to be. So let's discuss next this value. So in order to understand what confidence interval we have, let's remind ourselves what this x upper bar or what this sample mean comes from. We assume, in our case, it comes from normal distribution. And normal distribution is the one that you see in front of you just now. It has a bell shape. And this bell shape is spreaded around the mean of 0. It is symmetric function, which on the right-hand side and the left-hand side has the same shape. So basically, a mirror image around the 0. So what does z = 1.96 represents? The z = 1.96 represents something that we call the critical value. And this critical value corresponds to 95% confidence. The easy way to work this out is by looking at the statistical table that's provided at the end of each and every statistical book. The way to read this table is, first of all, we have to understand what confidence interval you're working with. But we already said that we're working with 95% confidence. And second of all, you will have to find out what the critical value is. And in this case, the critical value is 1.96. One important point to remember now is that 95.6 correspond to 5%, which is spreaded between two tails, on the right-hand side and on the left-hand side. Therefore, on the right-hand side, you have 2.5%, and on the left-hand, you also have 2.5%. And both those things represent 5% in total. So as you've just seen before, the probability density function of normal distribution looks like that, where in the right-hand side and the left-hand side, you have the critical values. Those critical values will be important when you want to find out the confidence interval for your mean. As you remember, you are taking the mean of the sample that you took plus minus the unknown factor. But now we can give some content to the unknown factor. First of all, we would like to say something about the error that we're happy with. And second, we also like to say something about the standard deviation. So the standard deviation of this bit, because remember what we said. What we said about this mean, that this mean can be of mean 1, mean 2, mean 3, and so on, as many means as we have as the way that we sample, but of course, we cannot sample those things. So we have some sort of an idea about the overall standard deviation of those point estimates. And this will be represented by the standard deviation, which looks like that. So the actual standard deviation of the sample over the sample size, square root because we're talking about standard deviation rather than variance, times, Times the critical value, which is, in our case, was 1.96, which correspond exactly to 95% confidence. I know this looks a bit confusing, but try to imagine it this way. If somebody asks you a question, what will be the actual mean of any population that you have in mind? So for instance, you have to take an exercise and you say, okay, what will be the mean of your exercises? You say, okay, out of ten questions, I can score an average 8 question plus minus 1. So to translate it to here, out of ten questions that you can do, eight, you are pretty much certain. So based on the sample that you have done, eight question you scored correctly, and plus minus bit will be this bit. So in normal language, it will say the average that you have from your samples, from your exercise that you've done, plus minus some value. And this value will be corresponding to how confident you are in what you're saying. Whether it will be a plus minus 2 questions, which will determine whether you have ten questions scored properly or six questions scored properly or plus, minus 1 question, so it will be whether you score nine question properly or you score seven question properly. This will determine, say, you mark the real mark in the examination. So again, let's rephrase it. What will be your examination mark, the actual examination mark? You'll say, okay, it's based on the past exam papers that you've done. You scored eight out of ten most of the time. But sometimes you got nine, sometimes you got seven, or sometimes you got ten and sometimes you got six. It will depend on how confident you want to be in your result. [MUSIC]
