Lecture 29

[Music] hello friend welcome to the course multivar procedure with r so you can recall that in the last lecture we had talked about the estimation of parameters and we had uh uh seen that how to use the method of Maximum likelihood for estimating the unknown parameters which can be univariate or which can be multivariate now you have seen that when we try to obtain the estimates of the parameter or the value of the parameter Theta based on any sample then as the sample will change the value of the parameter will also change how for example we had obtain in the case of normal distribution that the maximum likel estimator of mean is sample mean so that means you are going to collect a sample and then you are trying to find out the mean of those observation now definitely you can once you take another sample the sample values are going to be changed and then the sample will also change so when you try to draw different samples the sample values will be changing now how to ensure that which value is correct or which value is the correct value of the Theta or which value is representing the value of theta correctly the representative value of theta is the value of theta which is prevailing in the population not in the sample so one idea is that if uh there is not much difference between the values of uh parameter which are obtained from different samples possibly we can say that okay uh they are okay and they are representing the good value for example suppose if I make here a claim that my age is 5 years will you believe on it certainly no and if I say my age is 20 years you will say no it is not like this but if I say my age is suppose 55 years then possibly you cannot say no but the question is you don't need to know my age but my question is that when I said 5 years 15 years 20 years then you said no but as soon as I said 55 years you could not say no so what is happening that as soon as I say my age is suppose 5 years then you try to compare my structure my face etc etc and you try to compare it with someone who is already 5 years old or you try to compare it with the data which you have stored in your mind about the 5-year-old child and there is a huge difference between the two values and that is why you say yeah I'm not 5 years old but when I say that I am 55 years old then you try to compare my data with someone who is 55 years older maybe in your family Maybe your friend Circle Etc and then you say yeah my body structure my my eyes my hands my skin hairs on my head Etc they resemble more with a 55 year old person and that is why you believe on my statement so if you try to see this whole process was believing or not on a statement and this belief or the decision to believe or not is based on certain data I say a statement you try to create another statement in your mind and you try to compare it on the basis of some data set and then whatever is the outcome based on that you try to say whether I am correct or you are correct if I uh correct then you cannot be correct if I am not correct then you are correct so what is happening in between that how you are trying to manipulate this data inside your mind so that you can come to a conclusion that uh whether I am right or you are right this is the job which is done by test of hypothesis what is hypothesis hypothesis is only a statement so whenever we are dealing in statistics all our statistical procedures they are based on the random sample the random samples vary and consequently the values of those parameters meters values of those statistics they also vary and we want to test whether the values which you have obtained on the basis of given sample of data are they Dependable or not on the basis of certain criteria so now this test of hypothesis can be done on a univariate random variable on a multivariate random Vector also means including the multivariate random Matrix also so now from this lecture and in the next couple of lecture we are going to talk about the test of hypothesis and confidence interval estimation why confidence interval estimation that you will see that uh the results of test of hypothesis and confidence interval estimation they are inter related once you know the result of test of hypothesis then you can also get the result about confidence interval estimation or by looking at the result of the confidence interval estimation you can also take the conclusion about the test of hypothesis So based on different types of situation we have different types of test of hypothesis but uh surely as I say always my objective is not here to teach you the test of hypothesis and confidence interval estimation my objective is that I want to give you here the sufficient knowledge about these topic so that when we are trying to go further and when we try to use that concept of test of hypothesis and confidence interval estimation in our multivariate procedure then you understand it so with this objective in this lecture I am going to give you here a brief overview about this procedure that how do we conduct the test of hypothesis and in the next lecture we will take specific examples to conduct the test of hypothesis particularly about the mean in the normal population and the mean Vector in the multivari normal population so let us begin our lecture and try to understand about the test of hypothesis so in this lecture now we are going to talk about the basic of test of hypothesis some definition some terminologies and how do we conduct it right so as I explain you that we have noticed that when we try to draw different types of sample they will give us the different values of estimates and the value of these estimates may or may not be the same as of the unknown value of the parameters in the population right so this difference in the estimated value may be due to some random variation or there may be some assignable cost right for example in the example which I just took about my age when I said my age is 5 years you can say that okay there is a assignable call there is a reason and you cannot believe on my statement but uh when I said 55 years can you accepted it on the other hand if I say okay my age is 54 years 6 months then once again you cannot uh okay that it is not my correct age but the difference between 55 years and 54 years 6 months or 55 years 6 month is only a matter of six month which hardly makes any difference on anything so this uh variation can be caused it can be said as if this is a random variation right so the value which we want to know is unknown that is the unknown parameter so we want to check if the estimated value differs from some other reasonable value or not so if I try to it in very simple words then we would like to test the closeness of the estimated value with some hypothetical value what is this hypothetical value suppose in the same example of my age as soon as I say or make a statement that my age is 5 years you always try to assume a value inside your mind that if there is someone whose body structure looks like as of mine the age cannot be 5 years or the age should be more than 5 years or the age could be close to 54 years 55 years 50 years and so on so this is hypothetical value which is existing inside your mind so what are we trying to do we try to find out the difference between the estimated value and the hypothetical value and if this difference comes out to be less then we can expect that this is going to be accepted and we would say that there is not much significant difference between the two values right for example and I say my age is 54 years 6 months and and 55 years then you say there there is not much significant difference between the two values right and if this difference increases then we can expect to reject it and we would say that there is a significant difference between the two values for example if I make a statement my age is 5 years and you make a statement that my age is 55 years then the difference is pretty high and we see that okay there is a significant difference between the two values but the question now here is how to decide scientifically that the difference between the two values is significant or not right so we try to take the help of this the tool or the statistical technique of test of hypothesis so what's a hypothesis a statistical hypothesis is usually a statement about a set of parameters of a population distribution it is called a hypothesis because it is not known whether or not it is true for example if I say uh my age is 52 years now whether my statement is true or not that you don't know so that is why it's a hypothesis and a primary problem in such a case is to develop a procedure for determining whether or not the values of a random sample from this population are consistent with the hypothesis right because as soon as I say my age is 55 years you try to do some calculations in your mind and you try to get an answer yes my age cannot be 55 years or my age can be 55 years because it depends on certain body characteristics so what is that procedure so this is a challenge right so basically when we try to develop such a procedure then we have to make a rule right and this rule is going to be dependent on the random sample what we are observing from a given population so a test of a statistical hypothesis is a rule or procedure for deciding whether to reject the hypothesis or not right now in case if I try to take the example of a normally distributed population which has got an unknown mean value suppose here mu and some known variance here one now in this case variance is known so I'm not bothered about it but the mean is unknown so we would like to know its value on the basis of the given sample of data so now if I try to create here a statement that mu is less than 17 right that this mean it is a statal hypothesis and we could try to test by observing a random sample from this population whether mu is less than 177 or not and if the random sample is deemed to be consistent with the hypothesis under consideration then we say that the hypothesis has been accepted right for example if I say my age is 55 years and you say my age is 55 years 2 months there is not much difference between that two values and who can say that okay there is so my value is accepted and in case not accepted then we say that this hypothesis has been rejected so accepted and rejected these are the two terminologies that we use to decide the outcome of a test on the basis of given sample of data now this test of hypothesis can be conducted in different scenarios based on different types of sample so when we are working only with one sample to conclude about a value the problem is termed as one sample problem for example if I try to take a sample from a class to determine that that what is the age of the student in that class so I can have a sample based on that I can compute the sample mean and then I will conduct the test of hypothesis so there is only one sample and we want to make a conclusion only about the parameter based on one sample so this is one sample problem now on the other hand if I try to compare the average ages of two different classes then what will I do I will try to take one sample from class number one and say another sample from class number two and then I will try to compare their ages on the basis of given sample of data so now in this case particularly we are observing two samples to make a comparison and a conclusion about the value then this type of problem is termed as two sample problem and if I try to extend it when we are working with more than two samples to make a comparison and conclusion about a value the problem is termed as multivariate sample problem right for example if I say a school has 10 section in say class 10 so now there are 10 sections so now in an exam suppose I want to compare the average marks of the students obtained in every section so now there will be section one section two section 10 so we will try to draw here 10 different sample from each of the section and then we will have here a problem which is multivariate sample problem where we will be comparing the means of 10 samples right so now the question comes here how to construct that the decision rule so let me try to give you here uh this brief background that when we construct the test then we simply try to partition the sample space of X1 X2 xn into two disjoint subsets for example say CN C Star right suppose this is my here this sample space and I try to divide this into set C and C star and then we try to obtain here a sample X1 X2 xn and then we try to compute H 6 based on on that we try to take a decision that X1 X2 xn is belonging to suppose here C then we suppose we decide that if my sample values are belonging to the Region C then we will reject the H not H not is a hypothesis which we call as null hypothesis that I will try to discuss soon right and on the other hand in case if I say that X1 X2 xn ask us that that this point belong to C star that mean this region here then then we can say that we shall accept the null hypothesis H not right so based on this classification this C here is called as critical region or the region of rejection that means the sample values are going to lie in this region and we are going to reject the hypothesis on the other hand uh this region here C star this is call here as a acceptance region or the region of acceptance that means if the sample values are going to lie in this region we are going to accept the hypothesis and in order to avoid any confusion ambiguity we say that there is no common region between C and C Star right they are his joint there is no region where I can say that X1 X2 X1 will belong to C as well as C Star right now the question comes here what is the statistical hypothesis in terms of the parameters right so as I said that hypothesis is a statement about the parameter and this is Tal hypothesis has two part which is called here null hypothesis and alternative hypothesis so what does this mean suppose I try to take the same example that I try to make a statement that my age is 20 years and now you have to test it on the basis of sample of data whether my statement is correct or not in order to compare it you have to create one more statement now you create a statement here my age is suppose 55 years so I want to test my statement that my age is 20 years in comparison to your statement that my age is 55 years right so that is why I'm saying that here we have two parts one is here uh called as null hypothesis which is denoted by H not and this is a statement which is to be tested or that we want to test and other part is alternative hypothesis which is denoted as here H1 or say h and it is just opposite to the null hypothesis and this is a statement against which the null hypothesis is tested right both this null and alternative hypothesis they are mutually exclusive and only one of them can be true at a time either hot can be true or H1 can be true if hot is true then H1 cannot be true and if H1 is true then s not cannot be true right and whenever we are trying to test any hypothesis then we have to take a decision now this decision can have different situation and different out so let us try to understand it suppose in the actual situation there are two option that the null hypothesis can be true or the null hypothesis can be false now when you try to talk about the decisions what are the possible decisions one possible decision is that we try to accept the H not or we try to do not accept that is reject H not so these are the only two option whether you're trying to accept or not so now if you try to see what are the possible decisions so let me call it say here box number one box number here two in this way box number here three and box number here four right and let us try to analyze the decision each of the box so let us try to concentrate on box number one so when I'm saying that H not is true and we are going to accept it absolutely no issue there is no error that is what we want to do now you come to box number two we are trying to say that s not is false and we are going to reject it like is here like this right I can say here like this in that case absolutely there is no issue so now I come to here now we come to box number here three so you can see here s not is here false and we are saying that it's not is accepted that means we are trying to make a mistake on the other hand if you come to here box number here four four you are trying to say that hn is true and you are not accepting it so that means we are trying to make here mistake so you can see here in box number three and box number four we are trying to make two possible mistakes in making a correct decisions so these are two types of Errors which are happening in a decision making so the error which is happening in box number four that H not is true and we are not accepting it this is called as type one error and the error which is happening in box number three that s not is false and we are accepting it this is called as type two error now the question here is how to say decide the magnitude about this error so we try to compute the probability so the probability of type one error this is called here a Alpha and this is also called as the level of significance and this is the size of the critical region what is critical reason if you try to see in this slide we have talked about the critical reason here that this is the reason where our data lies and we reject the hypothesis or reject the null hypothesis right now similarly you come to here box number three so box number three is also making type two error and we want to quantify it so we try to compute the probability of making type two error and this is indicated by here like this so the type 1 error is indicated by Alpha and the type two error is indicated by the beta or more precisely the probability of type 1 error is indicated by Alpha and the probability of type two error is indicated by Beta right so these are essentially the terminologies that we try to use in the test of hypothesis similarly if you try to look into the box number here one the probability of no error is here 1 minus Alpha and and in box number two the probability of making no error is 1 minus beta but anyway uh we we will try to concentrate or work with the type 1 error and type two error only but definitely I'm not going to give you here more details because when we try to develop a decision rule we have to develop a decision rule in such a way such that the probabilities of these two errors are minimum and now minimizing both the errors is not possible so we try to follow the rule which is given by the neon pearon LMA which is based on the basic assumption that uh type 1 error is supposed to be to be more serious then typee two error right so what we try to do here that whenever we try to develop the decision rule then following the neon Pearson LMA we try to fix Alpha and we try to minimize beta that is the type two error so that is the basic fundamental rule so I'm just giving you an idea and if you try to read from from from any book or any types of standard test statistics that we are going to use here they have been found using the name Pi LMA and they are based on the assumption that type one error is more serious than the uh type two error right so now I will just try to give you here a formal definition of different terminologies so what is the meaning of accepting H not this means that the difference between sample mean and hypothetical population mean is not significant ific or whatsoever is the different this is coming because of the sampling fluctuations only that means the there is no assignable cause it is only the random variation because of which the different values of estimator from different sample based on different sample they are varying right now another Concept in the test of hypothesis is about P values right so as I told you that when we try to take a decision whether to accept or not the null hypothesis then we try to decide the critical reason and then using our test we try to decide where it is going whether to the critical regon that is the acceptance region of rejection or region of acceptance but when we are trying to work in the software the software try to compute uh the test statistics and they try to report the critical values using the concept of P values right and it is possible that instead of using the concept of the CC star that is the critical region or acceptance region we can also use the P value concept to make the decision of the test right so P value is essentially the estimated probability of rejecting the H not that is the null hypothesis right so based on that we try to make our decision Rule and now um when we are trying to test the hypothesis then we have two types of uh test which is one-sided and two-sided and in one-sided we have left till or say right tail test and two-sided test mean left tail and right tail both right so we try to compute the P value for a given statistics TX in the case of two sided hypothesis as probability of this absolute value of T is greater than TX under H when H not is true right this means when H not is two and for one-sided we try to compute the P value as the probability of T greater than equal to TX when s not is true or t less than equal to TX when s not is two for right hand side and left hand side hypothesis what is called as right tail test or the left tail test right I'm not going to give you all this detail but I will try to show you in the examples so right and the decision rule is extremely simple the P value will be provided by the software and the level of significance that is the size of the critical reason that is the value of a probability Alpha this is decided by the experiment that can be 5% level of significance 1% level of significance or whatever is more suitable for the given experiment now so you have the value of alpha you have computed the value of P now if if P value comes out to be is smaller than the value of alpha that is the significance level then we reject the null hypothesis right and significance level is the probability of type one error Alpha as we said earlier right so now what are the steps when we try to conduct the test of hypothesis so in order to test the null hypothesis we require a good statistical test which is uh developed by fixing the type one error and minimizing the type two error and this is based on the neon pearon LMA right right so these are the steps so first we try to define the distributional Assumption for the random variable of interest and you specify them in terms of population parameter for example that X1 X2 xn they are coming from normal population with the MU and uh andma square based on that we can say that whether mean mu is known or unknown or Sigma square is known or unknown like this then we try to formulate the null hypothesis and alternative hypothesis right then we have to fix the level of significance that is a type one error and then we try to develop a test which is equivalent to developing a test statistics right now this text statistic is constructed is computed on the basis of the given given set of data right if my TX which is a function of tx1 X2 xn it is now computed on the basis of random sample and its value is calculated and then we try to construct the critical reason k for the statistics T that is the region where if the value of test statistics fals for a given set of data then we will decide that s not is rejected right and then we try to compute this value of TX for the given set of data and then this test statistics is follows actually a probability distribution and that is the job of statisticians to find out and uh then we try to find out the corresponding value of the test statistics based on the probability distribution from the table of say for example if the test statistics is follow a Ki Square distribution then we try to find out the critical values from the tables of Ki Square probability if the test statistics is following a t distribution then we try to find out the value of the critical region or the critical values based on the probabilities of T distribution and so on and then we try to compare these two values the values which we have obtained on the given set of data and the value which we we have obtained from the probabilities based on the distribution of the test statistics and the rule is like this if the value of the TX that is the value of test statistic based on the sample of data falls into the critical region then H not is rejected and automatically the alternative hypothesis is then accepted on the other hand if the value of the statistics for the given set of data TX fall outside the critical region then H not is not rejected right and H not is accepted right and if you are working with the software then you have to look at the P value and if P value comes out to be smaller than Alpha then H not is rejected the null hypothesis is rejected right and yeah there is a very good relationship ship between the decision based on test of hypothesis and the confidence interval right and this is also used in making the right decision sometime you will see in the software they will give you only the confidence interval and based on that you have to decide whether a particular type of test of hypothesis is getting accepted or not but the rule is very simple if hn is rejected at a significance level Alpha then there exist a 100 1 minus Alpha per confidence in interal which yields the same conclusion as the test right suppose my null hypothesis is s not Theta equal to Theta not Theta can be some parameter like as Mu and we want to suppose test s not mu is equal to say 20 right so Theta 0 is some known value and uh we try to create the confidence interval for Theta now if the confidence interval of theta does not contain the value Theta KN Theta KN is is your known value right right then H not is rejected so this is how we try to take the same conclusion that we try to obtain on the basis of the test of hypothesis using this confidence interval right so now we come to an end to this lecture and uh you can see here that I have uh given you some details that how are we going to conduct the test of hypothesis and as as I said that uh that that I have my objective was only to give you a brief overview right there are many Concepts which are related to the test of hypothesis I have not given it here because it is not my objective here because I believe that you are aware of the test of hypothesis concept and I want to extend it to a multivariate case so my objective was here to give you uh some knowledge so that me and you are at the same level and you know that what exactly you have to know in order to understand the lectures in the further course so my request to to you will be here that at least you try to brush up your uh Concepts about uh the test of hypothesis and in the next lecture we are going to revise the test of hypothesis related to the mean in a univariate casee in the case of normal distribution and how to conduct it in a our software that will be another issue so I will try to start with the univariate case and then I will try to bring bring you to the multivariate case as well as when we have more than one random variables for example testing of mean from a single population from two population and then more than two population then conducting the test of hypothesis on a univariate random vector and then to a multivariate random Vector so that is why in order to prepare my background to explain you about the multivariate case it is important for me to First give you some information about the univariate case so that uh the understanding become easier for you so you try to practice it and I will see you in the next lecture with more details till then [Music] goodbye [Music] [Music] [Music]