 dealing with materials data cores. For past few sessions, we are trying to learn about the population through several statistical methods. Now what we are going to work with is called hypothesis testing. What we did in the past was our endeavor was to learn about the population from a small sample. So, what we did, we did the estimation of population parameter. We assumed that the data comes from a population with a certain distribution function and then that distribution function had certain parameters which were unknown. So, if we know the parameters, we know the population itself. So, we learned several techniques of estimation of distribution parameters which is known as parametric estimations. We had point estimators and we added the accuracy into it by having an interval estimate. What we want to do now is what is called hypothesis testing. What does it mean by hypothesis testing? Well, if you have the data and you know that it comes from a certain population and you also know that that population has a certain mean. Let us take an example. Suppose a person wants to buy or a company wants to buy a certain kind of an alloy and it is interested in having that the alloy should have some good property of yield strength. The company tells him that the supplier tells them that yes, our alloy has a yield strength value y. How does the buyer know, the customer know that truly what they are getting as a consignment has the yield strength y. So, what they would like to do is take a sample from the consignment, test their yield strength and then check or test if this yield strength is same as y. This is called hypothesis testing. So, in this today's session we are going to look into five basic elements of hypothesis testing. The first one is simple and composite hypothesis. The second one is a critical region resulting possible errors while you do hypothesis testing. What is called an alternate hypothesis and classical approach to hypothesis testing. Consider we as I said let us have a population with a distribution function f theta where theta is the parameter and it is unknown. If we know theta, we know the distribution, we know the population then there is no question of doing any statistics but we do not know theta. Let theta not be a specific known number or unknown value. So, as I said in the previous example, if what I said is that the company claimed that my our material has yield strength y, then this y is theta not. Then we would like to know whether the unknown parameter theta is truly theta not. In statistical parlance we say that we want to test the null hypothesis that theta is equal to theta not and it is denoted by a null hypothesis. So, it is called H0 is theta is equal to theta not. This null hypothesis there are two kinds. If the null hypothesis of the kind that H0 says that theta is equal to theta not then if you actually put theta equal to theta not then it completely it would specify the population distribution. Again you know the whole population what the population behaves like. So, if we say that if null hypothesis is true and if it completely specifies the population it is called a simple null hypothesis. While if you take a null hypothesis which says that for example company may say that my yield strength is at the most so much. It means that your null hypothesis is that theta is less than or equal to theta not and if this is true it does not completely specify the distribution and therefore the population and therefore it is called a composite null hypothesis. Next what we would like to do is we want to find a critical region. Now here what really we would like to do is something like this. We have a sample space. Please do not forget this old terms. We have a sample space which is a result of all the experiments that you conduct. All possible results of the experiment that you conduct. So, here for example if we are continuing with the yield strength then all possible yield strength that it can have will sit in this. Now x1, x2, x3, xn is a random sample. So, it will give you some value of this is your sample x1, x2, x3, xn. It is a sum set within the sample set. We would like to define a region within the sample space which I called C in such a way that if this sample belongs to this critical region I can confidently say that reject the null hypothesis. I would say that reject the null hypothesis. I am defining a critical region to reject the null hypothesis. If it is not in this and if it is sitting outside I say that okay I do not have enough evidence. I cannot reject the null hypothesis or I accept the null hypothesis. Why are we going on the rejection? In day to day life you see to accept anything requires a lot of justification, lot of proofs and lot of convincing. Rejecting anything is much easier. So, it is true even in terms of statistics and mathematics. Therefore, we try to find a region which will reject the null hypothesis. So, here in this region we are going to reject the null hypothesis. Where else we are going to accept? Generally in statistics we do not say we accept it. We only say that we do not have sufficient reasons to reject it. Because there is always a doubt had my sample been larger had it been another sample possibly it would have been rejected. And therefore we always say that we did not have sufficient evidence to say that the null hypothesis can be rejected. So, this is how we define a critical region. Now when you do such a hypothesis testing there are two kinds of errors that we can commit. Let us look at it in a systematic way. The reality says that null hypothesis is true or it says that null hypothesis is not true. Based on the hypothesis testing finding a critical region we can take a decision accept null hypothesis or reject null hypothesis. If you accept the null hypothesis which is true there is nothing to worry. Similarly if you reject the null hypothesis when it is not true there is nothing to worry you have done a right thing. But if we accept the null hypothesis when it is not true it is called type 2 error and the another error is when you reject the null hypothesis when it is true it is called a type 1 error. These days with respect to medical testing it is very common to come across two terms. One is called false negative and the other is called false positive. Type 1 error refers to the false negative. It is a negative answer wrongly given. It is a negative answer which is wrongly given. False positive is that it is a positive answer. You accept the null hypothesis but it is wrongly given. Therefore it is called false positive the type 2 error and type 1 error is called false negative. At this point I would also like to bring another issue which is very common with the machine learning technique which is getting more and more prevalent in all subject areas of science and technology. We use something called neural networks. It is a in a way a very sophisticated curve fitting technique where we have a very highly non-linear curve which we have a neural network as a tool to estimate it and what we generally to do? We generally have a set which we called a test set or the fitting set and then we have a test set. So, we take one set of observations. We do the iterations and we find a neural network model and then we have a test set in which we put that values in it and see that it truly fits within the boundaries or not. We have something like a least squares error there and then we try to see that even the test set falls within the boundaries of the specified minimum error limits. In all this process it is something which is not done is finding the type 1 error and type 2 error. That effort should also it is not very difficult to do not a part of this particular course. So, it is not being covered, but it is just to bring it to your notice that in the euphoria of applying machine learning and therefore using techniques such as neural network please remember before using this heuristic methods also devise a method to identify the two types of error which can occur in any decision making procedure. Any decision, this is a decision making procedure and this is kind of a error table for a decision table. So, in such situations even when you use any heuristic method, heuristic method is like neural networks or any other method. This two kinds of errors should also be tried to be evaluated. Anyway let us come back to our present context of hypothesis testing. These two errors frankly we would like to minimize both the errors so that we do not commit any great error. Mathematically it is not possible. The classical approach to hypothesis testing says that fix some value alpha at the minimum possible level and then set up a test so that your probability of type 1 error is smaller than alpha and then you study how much is the type 2 error. This is the classical approach to hypothesis testing and with this we have two values now. As you can see we are going to fix the some value below which the type 1 error should be and we will study the type 2 error accordingly later. So, in this case there are two important aspects come into picture. What we are trying to say is that you make probability of type 1 error smaller than alpha. It means that you are going to reject the null hypothesis when it is true that probability you would like to make it less than alpha or it is same as saying that your sample will lie in the critical region when the null hypothesis is true that probability is also less than alpha. Such an alpha is called a significance level of the test. The alpha level that you have fixed it is called a significance level of the test and 1 minus alpha is called the confidence level of the test. Generally in practice you would have heard it and you might have used it in the past. This alpha is kept at 1%, 5% or 10% level. Please remember this is kind of arbitrary. No specific meaning is attached to why 10% and why 1% and why 5%. But important thing is before you do anything in the testing of hypothesis the level of significance alpha is to be fixed. It should not get affected by looking at your further analysis data, analysis results. It should be fixed up right in the beginning. 1 minus alpha is called the confidence level. So if your alpha is 5% then you have a 95% confidence level of the test. Look at the type 2 error. Probability of type 2 error is generally denoted by beta. It means that you are going to accept the null hypothesis when it is not true. It is same as reject the null hypothesis when hypothesis is not true is 1 minus beta and therefore we say that 1 minus beta is the power of the test that you are you are able to reject the hypothesis when it is not true that is called the power of the test. Now when I say that okay my hypothesis is that theta is equal to theta not. Now I define a critical region that where hypothesis is not true. So when theta is not equal to theta not there are many of alternatives. It could be that theta is just not equal to theta not it is neither greater nor less or we can say that theta is not equal to theta not but actually theta is smaller than theta not or I can say that theta is not equal to theta not but actually theta is greater than theta not. These are the called alternate hypothesis. The notation says that just as we write null hypothesis as H0 or H0 alternate is written as HA or H1, H2, H3 etc. Now what is I would like to give you a very basic classical approach to this hypothesis testing. It covers 6 steps and I would like to suggest that whenever you do or solve any problem please go one by one through each step because as you will see in future or when you do it yourself you will realize that it kind of tends to get confusing in the middle. So it is a good practice to follow this classical approach of 6 steps step by step. So first step is fix the significance level of alpha very important before you do anything significance level of alpha should be fixed. Then clearly state what is your null hypothesis and what is your alternate hypothesis? Third choose an appropriate estimator of theta using the data X1, X2, X3, Xn let us call it D a function D of X1, X2, X3, Xn or we call it DX vector. Then we define a critical region using this D where H0 is rejected. Then we calculate the probability of critical region when H0 is accepted. We compare it with alpha and we determine the exact nature of C and then we know how to give the decision. Once you know this probability, you know the nature of C, you know the probability and then it is simply says that if it is less than alpha or equal to alpha you are going to reject it. Otherwise you do not have sufficient evidence to reject it. So you accept it. So let us review this quickly. We saw the necessity of hypothesis testing. That is there are situations where the estimation of a population parameter is not just sufficient. You have to check or you have to test whether it equals to the value which is pre-decided or already given to you, whether it equates to that, it is smaller than that, it is greater than that whatever. If it defines a completely the your population form, your population distribution by the null hypothesis, we call it a simple null hypothesis. If it does not, for example, null hypothesis could be that theta is less than theta naught. In that case it does not define the population completely at the end of testing of null hypothesis and therefore it is called a composite hypothesis. Then we said that it is much easier if we can find out a region in the sample space where if our observed sample falls then we can say that reject the null hypothesis. That is called a critical region. Then we said that well what can happen? Two types of errors can occur. The first type of error is when you reject the null hypothesis when in reality it is true. It is called false negative. The other type of error is you accept the null hypothesis when it is not true. Then it is called false negative. These, did I say false, now it will be false positive and then we talked about what are the alternate? I am finding, I am proceeding the way that first I have a region where I reject the null hypothesis. So, in that case I must know if I reject the null hypothesis what is the alternative? That alternative I call it alternative hypothesis and then we have defined six steps of classical approach. In this summary I would like to emphasize once again two things. The type 1 and type 2 errors are of extreme importance. They drive the whole classical approach and they should be driving even the new fresh approach that we take in variety of problems of decision making. These two errors should be taken into account that gives the strength to your test. Second thing I would like to say is that the type 1 and type 2 error are a part of decision making procedure and therefore they should be applied as and when whenever a decision procedure is taking place. However once again I repeat in this particular course we are going to restrict the meaning and interpretation of type 1 and type 2 error only up to the hypothesis testing in the classical statistical manner. Thank you.