Lecture 32

[Music] hello friend welcome to the course multivariate procedure with r so you can recall that in the last two lectures we have considered the test of hypothesis for the mean from a normal population normal mu Sigma Square under two uh situations when the population variance Sigma square is known and when it is unknown and we had considered there one sample test that means we had only one variable on which we are trying to obtain the data in practice there are many situation where we want to compare for example I want to compare whether the population mean of two different population is the same or not on the basis of given sample of data for example if I say there is some medicine and Suppose there are two medicines to control the body temperature now both these medicines are given to uh to a group of patient suppose I have group one to whom I have given the medicine number one I have group number two where I have given the patient uh the medicine number two to the patients now I want to see whether these two medicine they have a different effect or they have the same effect or also whether medicine one is giving a better outcome in the sense that it can control the body temperature for a it can maintain the or it can bring down the body temperature in a uh lower time Etc different questions can Beed about the comparison of the different uh type of body temperature in the two groups of the patient so now suppose if I want to address these types of issue in simple way I can say I want to know whether medicine one is more effective or medicine two is more effective or whether medicine one and Medicine two both are equally effective or they are they have different effects so in this case we have two samples so under this type of situation when we want to make a comparison using the test of hypothesis then how to do it so I have here two condition that I have here two samples which are drawn from two different normal populations and these populations are differing with respect to the mean yes they can differ with respect to the variance also but in order to understand this concept over here I'm considering that they are differing with respect to the mean mean only and they have got the same population variance now this population variance which is common to both the population in the sense that they are they have got the same value that can be known or that can be unknown so when it is known then in the case of one sample test we have uh used the gaus test and we had used the command gaus dos test in our software and when this Sigma Square was unknown then we had computed it we had estimated it on the basis of given sample of data and we had used the T do test command that was built in in the base package if you recall at that time I had told you that in these commands I had two options X comma Y where I had asked you to use y equal to null because we were considering only the one sample test but now we are in a position where we are going to use both X and Y so in this lecture we will consider both type of test of H hypothesis when Sigma square is known and when Sigma square is unknown and we would like to compare the population means of two normal population so let us begin our lecture and try to understand so now in this lecture we are going to consider the test for mean in the two samples with the univariate data right so first we try to consider the case when the population variances are known so we have got here two populations which are differing with respect to the means and they have got variances but they are known to us right so we are interested in testing the hypothesis for the difference of mean difference of means means for example if I try to write H not mu1 equal to mu2 then this can also be written as H1 mu1 minus mu2 is equal to Z or in more general statement I can write in mu1 minus mu2 equal to Delta right so and both this sample they have been drawn independently from the two population which are normal mu1 Sigma 1 square and normal mu2 Sigma 2 squ right so now we have now these two samples from normal mu1 Sigma 1 a squ and normal mu2 sigma2 square and uh the first sample is of size N1 and it is drawn from a population whose mean is mu1 and variance is Sigma 1 squ which is known and then we try to compute or estimate the sample mean which is here X1 bar then we try to consider the second sample of size N2 which is drawn from a population which has population mean mu2 and population variance Sigma 2 s which is also known and then we try to estimate Mewtwo by the sample mean xar 2 and we wish to examine if two population mean mu1 and mu2 are the same or they are different right so now uh the null hypothesis is going to be H mu1 equal to mu2 right and then we have here three uh possibilities to construct the alternative hypothesis so one option is that I can consider the two-sided hypothesis that is H1 mu1 is not equal to mu2 right or other alternative is that I can consider one-sided hypothesis H1 mu1 is greater than mu2 which is your here right tail test or H1 mu1 is less than mu2 which is a left tail test and this two sided it is a two-tail test right now because population mean mu1 and mu2 are unknown then we try to use the statistics X1 bar - x by2 to make some conclusion about mu1 minus mu2 right so we are assuming here that Sigma 1 and sigma 2 both are known both the parent population are normal uh or I can assume that sample size N1 and N2 are large if you try to see these assumption are similar to the test or say is test that we have considered in the earlier lecture right now we know that sample mean X1 bar will be following a normal distribution with mean mu1 and variant Sigma 1 s upon N1 the second sample mean X bar2 will be following a normal distribution with mean mu2 and variance Sigma 2 s upon N2 So based on that I can write down the expected value of X1 barus X bar2 is mu1 minus mu2 and variance of X1 bar - x by2 is uh Sigma 1 s upon N1 + Sigma 2 sare upon N2 and the co-variance between xar 1 and xar 2 will be zero because they are drawn independently based on that we try to create here a test statistic see here z c so if you try to see here this hypothesis H mu1 = to mu2 I'm trying to consider here as say H1 H mu1 minus mu2 is equal zero right so now if you try to look at this Statistics say X1 bar - x 2 - 0 then the zero is coming here because of this zero right in case if that is something else say Delta then the zero will be converted to Delta so now I try to divide it by the standard deviation Sigma 1 sare upon N1 plus Sigma 2 sare upon N2 and a square root so this is the test statistics and when H not is true then this zc follows a normal 01 distribution and so we try to use the probabilities from the normal 01 distribution to compute the critical values and the P values right so now if you want to implement this test in the r software then as we have discussed earlier that we are going to use the uh command gaus do test which is available in the library compositions right compositional data analysis so this has to be installed first and then this has to be uploaded so this can be done by using the command install. packages and Library compositions and uh we have used it earlier in the one sample test case now the same command I'm going to use it in the this two sample case also so the command here will be gaus do test G here is in uppercase alphabets now you can see here this x is indicating the sample value from the population one so this is sample one and Y is the value which is in the sample two right then mean here is zero this zero is indicating this value H not mu1 - mu2 is equal to 0 this Zer is this zero here right and standard dation well it is here one but that you can give here any value right and uh yeah on the other hand I can also say that if if you are trying to consider H not mu1 minus mu2 is equal to some Delta then this zero will be replaced by this Delta value the alternative will remain the same as per your choice and the requirement as two-sided less or greater as we have done in the earlier Command right so yeah this is the same thing that I just explained you so now we try to consider an example and we try to implement it in the our software right uh suppose we have two samples which are about the gain in the weights in grams of Fes which are given two different types of dietes say call As A and B so ises are given two types of dietes A and B with an objective to understand whether which of the diet is going to increase more weight right so the increase in the weights of these fishes they are recorded here as a in the diet a as like this in grams and in diet B they are recorded here as say in grams and suppose the standard deviation is known to be here two right and we want to test if the two dietes differ significantly as regards their effect on the increase in the weight right this standard deviation two it is actually the value of Sigma 1 s upon N1 plus Sigma 2 s upon N2 right so I'm just assuming it to be no known so now if you try to see I try to give this data in two data Vector x a and XB right now I try to use here the command ga. test and the first data set XA from here second data set XB from here mean is equal to zero because H mu1 minus mu2 is equal to0 standard devation is equal to 2 this is known this is given to us and alternative I'm trying to take say two-sided H or say H1 mu1 is not equal to mu2 and this is here the outcome you can see here so you can see here the data here is used here like this and T here comes out to be here minus 2 this is the value of the test statistics and P value comes out to be here 0. 9823 right so now the rule here is that if P value is equal to 0.0098 and suppose you have chosen the level of significance Alpha to be 5% level of significance then the rule here is reject H if p is less than Alpha so P here is 0.0 098 and Alpha is at 0.05 so what do you think this is correct so in this case this H mu1 = to mu2 is rejected that means gaining the weights from this D A and diet B they are not equal or alternatively H1 mu1 is not equal to mu2 is accepted that is correct that both the dietes are giving on an average difference increase in the gain in the weights right okay so now this is here the screenshot and now I try to show you this thing in the AR console also so that you can be more confident about it right so first let me try to load this package although so now this here is Library composition as I said I already had uh uh uploaded this thing in my computer now I try to give here the the data here and we try to execute this command here so you can see here this is here like this now if you want to change the alternative you can make it here less or if you want to make it here greater greater you can see here here like this right so you can see here that the P values in the three alternative that is coming out to be different right okay so now let me come back to my here slides and try to consider the other test so now we move forward and our problem or the statement of problem is the same that we want toess the hypothesis about the equality of two means from two different populations which are the normal population whose means are different and their variances are not known so up to now we have assumed that Sigma 1 and sigma 2 both are known and for that we have used the statistic zc which used to follow not Al 01 under H not now we are going to assume that both Sigma 1 and sigma 2 or Sigma 1 square and sigma 2 sare they are unknown and in this case we are going to use the T distribution that means whatever be our statistic that is going to follow a t distribution now this is a similar case if you try to recall that in the last lecture we have considered this setup in a single sample and uh we have used the command t. test so in order to now extend it to two sample the same assumptions are going to continue but then there is small variation we are assuming that both populations are normal that mean the samples are drawn from a normal population and we are assuming that both Sigma 1 and sigma 2 are equivalent to Sigma 1 square or and sigma 2 square they are unknown but they are equal there can be other case also that they are not equal but the test statistics for that case is different than what we are going to do here that is called as Fisher bar problem and that is handled in a different way and that is applicable when uh Sigma 1 square and sigma 2 square they are unknown as well as unequal right and in the two sample case also if the sample size is large enough say greater than equal to 30 then we can approximate the T probabilities by the normal 01 also and the reason is the same what we have discussed earlier in the case of one sample test so now we have here the same set of hypothesis that my null hypothesis is H mu1 equal to mu2 and my alternative can be mu1 not equal to mu2 mu1 is greater than mu2 or H1 mu1 is less than mu2 now we are observing a sample of size M from normal mu1 Sigma square and another sample of size small n y1 Y2 YN from normal me2 Sigma 2 s right and both mu1 and mu2 are unknown so we try to estimate mu1 by say xar and mu2 by y bar so I can uh use this xar minus y bar to make conclusion about mu1 minus mu2 why because if you try to see this null hypothesis can also be written as H not mu1 - mu2 is equal to Z so now we have here both Sigma 1 and sigma 2 both are unknown but they are assumed to be equal so now we try to create here the test statistics so we know that xar will follow a normal distribution with mean mu1 and V Sigma 1 s upon M the sample mean Y Bar will follow a normal distribution with mean mu2 and variance Sigma 2 s upon n and expected value of xar minus y bar will be equal to mu1 minus mu2 and now we try to compute or estimate the VAR as cooled variance how to do it we try to estimate the variance Sigma Square by this expression 1 upon m + n minus 2 and then inside the backet summation x i - xar S Plus summation y i - Y Bar s right now based on that we try to construct here the test statistics which is here like this DC so once again if you try to see I had the null hypothesis mu1 minus mu2 equal to Z so we try to write down TC here as say xar minus y bar minus 0 this zero is coming from here right and divided by the standard error which is here like this right so this is my T statistics TC which follows our T distribution with M plus and minus 2 degrees of freedom when H is true so in case if uh sample is large then we can use the normal distribution to get the critical values or when we are trying to use the software then we can use the P values to get the uh to get the conclusion or the decision about the test of hypothesis so now let me try to take here the same example about the diet a and diet B which are given to FES and we want to test whether the effect of these two dietes in increasing the weights of the fishes is the same or different all right so this is the same example the only difference is that earlier I had assumed the standard deviation to be known as to but now I don't know and I would like to estimate it on the basis of given sample of data so this is here the data on D A and D B which is stored in say two data Vector X and XB but now suppose the standard deviation is unknown right so we will estimate it from the sample and now we want to test the same hypothesis s not mu1 equal to mu2 versus S1 mu1 is not equal to mu2 now if you try to see the command here the command here is the same what we used head earlier exactly the same command now the only difference here is this now I am giving here x as x a and data on y in the one sample case it was null but now it is here given as here say XB that we already have understood in the earlier uh lecture and in the earlier command also the alternative here is two sided mu is equal to here Zer it is coming from here mu1 minus mu2 is equal 0 and paired is equal to false why because it is not a paired test because we are getting uh the samples independently to samples are independent if you recall if I in the earlier lecture I had explained you that under what type of condition the observations are going to be paired and then we try to use the paired T test right now the next option is variance do is equal to true so now you can see here I have assumed that the two samples are coming from two different normal population which are differing with respect to only means mu1 and mu2 but their variances are unknown but equal so now I have to give here true V are DOT equal that is variance equal is equal to true the level of significance Alpha here is uh suppose I take 5% to conf dot level will be 1us Alpha which is here 1 minus 05 which is 0.95 now if you try to see here this is the outcome so this is the two sample test data used here is XA and XB and this is here the value of T right and this is here the value of degrees of freedom m + nus 2 and this is here the P value alternative hypothesis is is two-sided so the two difference in the mean is not equal to zero right is H1 mu1 minus mu2 is not equal to Zer and the 95% confidence interval here is obtained like this right and the sample mean of X and Y they are obtained here like this 28 and 30 right so if you try to see that if I try to conduct it on the this R console also this is screen shot here which I would like to show you now right so if you try to see we already had stored the data on say XA and XB when we conducted the test when Sigma 1 s and sigma 2 s are known so XA is here like this XB here is like this and if you want to consider here this T Test you can see here this is here the outcome right so you can see here that it is the same outcome which I shown you on the our console now if you want to change here the alternative hypothesis suppose if I want to take it here Le than type left tail test so in the left tail test it is coming out to be here like this the confidence interval is changing the P value is changing and if I want to make it here for greater than means H1 is mu1 greater than mew2 so for that you can see here I simply have to change the alternative and you can see here that the T value will remain the same in all the cases the confidence interval is changing and the P value here is changing right right so now we come to an end to this lecture and uh you can see here that in this case what we have done whatever we have done in the last two lectures they have been extended from one sample to two samples but if you try to see the way they have worked that is the same the methodology is the same the only difference is coming because now we are using different test statistics when Sigma 1 square and sigma 2 square are known or in general when the variances are known then the test statistics is following a normal 01 distribution and when the variances are unknown then it is following a t distribution you have to be careful that when you are going for the case when uh variances are unknown then there are two possibilities that the two variances of the population they are equal or say unequal based on that you have to choose different statistics well when they are unequal uh we try to that problem is actually called as Fisher bam problem but now you see in the software you need not to worry you simply have to just uh change with where do equal is equal to false and you are done and after that you have to Simply look into the value of p and then you have to compare it with Alpha so that is the advantage that when you are trying to conduct the test of hypothesis in a software then many things become easy but it is important for you to understand that what is happening in the background that is why this theory is important so now uh we have couple of jobs first we consider one sample test then we consider then we extended a two sample test now would you like to extend it to a general setup when when you want to or would you like to compare more than two population means the next option will be you are considering here a univariate random variable but would you like to consider it in a multivariate data then whatever you have done one sample test two sample test Etc would you like to extend it to a multivariate setup yes that is our objective what we are going to achieve in the next lectures but in order to understand them it is very important that by this time you have uh completely understood the philosophy behind the one sample test two sample test and the different type type of conditions and what are the different methodologies as uh as well as steps which are involved in the conduct of test of hypothesis so for that I would request you you try to create two data set yourself in one data set you try to maintain lower variability and in other data set try to maintain higher variability and then try to see try to compare them what happens or try to take the two data set which is are of the similar nature similar values and try to compare them try to create different types of hypothesis try to interpret them the interpretation is very important but one thing I would like to explain you here before I leave that all these tests of hypothesis they are based on certain rules of the statistical inference whatever Z statistics T statistic either in one sample or two sample you have used they are derived using the Nan Pearson LMA and neon Pearson LMA gives us a methodology to derive such good test under the assumption that the null hypothesis has been constructed in such a way such that type 1 error is more serious than type two well both are serious type one and type two both but the way this Z test and T Test whatever we have considered and whatever we are going to consider in the further lectures also they are also constructed on the same assumption that the null hypothesis has been constructed in such a way such that type 1 error is considered to be more serious than the type two errors that mean the consequences arriving arising by the violation of the type one error are more serious in nature than the type two errors well the reason is that it is not possible mathematically to minimize both the errors simultaneously if you try to minimize one the other goes up and vice versa so that is why one is kept fixed the alpha is kept fixed at 5% 1% level and the second error is minimized in the best with the best level yes you can also uh also ask me what if if the type two error is more serious then yes you can derive different test statistics and then you can conduct the test of hypothesis but definitely this is not the objective in this course I'm not going to tell you how to do this thing so better is to use this existing test of off of hypothesis statistics and try to frame your null hypothesis in such a way such that type 1 errors is comes out to be more serious than the type two error so you try to practice it try to take some data try to expose them to the art software try to understand the theory behind these test of hypothesis from the books and I will see you in the next lecture till then [Music] [Music] [Music] goodbye [Music]