 तब कि आप बाख of this, this is the 2 sample SEO sample. उआप भाख of the 2 samples, what is the sample? जी बाख of the single sample of the code. अरे बाख of the next sample only if we haven 2 samples, तब आप पाख of the next sample only, how do we analyze it further? तब बाख बाख of the two main vectors, इप बाख of the two different diseases. तब ब बाख बाख of the 2 samples, और the next sample only, तब बाख of the two samples, बाख of the 2 sample SEO sample. तो i, which is equals to 1, और i, which is equals to 2. तन we know, the x bar 1 is distributed at multivariate normal, and mean vector mu1, and variance covariance matrix, तग्मा over n1. प्रोपल याप को यादे जब हमने, x के लिए जब को फाँईट की है, तो mean vector बहुत हमारे परस्ट, and variance covariance matrix, sigma over n तग्मा over n तग्मा. तुके आब हमने, सामपल वान, and सामपल तुके लेचे करे, तो, mean vector mu1 है, and variance covariance matrix, sigma over n तग्मा. तो, x bar 1 is distributed at multivariate normal, with mean vector mu1, and variance covariance matrix, sigma over n तग्मा. तो, similarly, second sample, the x bar 2 is also multivariate distributed at multivariate normal, with mean vector mu2, and variance covariance matrix, sigma over n तग्मा. An unbiased estimate of sigma is given as, this is the sp. What is the value of the sp? Sp stands for pooled variance. In the univariate case, you may have read the sp. So, we have the pooled variance, and we have the formula for the sp. This is for the first unit, and this is for the second unit. And 1 over n1 plus n2 minus 2, this is the total sp. Now, the property of multivariate normal distribution, as you know that, the x bar 1 minus x bar 2 is distributed multivariate normal, with mean 1 mu1 minus mu2, and this is the variance covariance matrix. And this constant value is multiplied with this. Then the hoteling t-square distribution of two samples is given as this. This hoteling t-square is further, we have to use the hoteling t-square for two samples. So, for the test, now we have to do the hypothesis test. For the hypothesis test, what you have? If, condition if, mu1 equals to mu2. For what? We did it for two samples, and for what? We are doing two-sided tests. Mu1 is equal to mu2 is true for delta-square equals to 0. Means mu1 minus mu2 equals to 0. Here we are testing. So, what you have? These test statistics are done. t-square, this is the two samples we are checking, two samples of the hoteling t-square. These are the test statistics. Now, you know that, we have to give it a conclusion. For the conclusion, reject and accept. So, we have this critical value. Here, you will find this critical value. And you know that, the t-square which is equal to the f, and the f we have, this is the degree of freedom. If it is one-sided, so, alpha f, alpha will be, and if it is two-sided, then we will use alpha by 2. And reject H naught. If we reject it, and reject H naught, if t-square greater than and equals to this factor. This f critical value or you have a stable value or you are multiplying it. So, we have just rearranged it. We have written f later. And the constant term which is multiplying from f, we have written it earlier. So, we can use this value as well. Both the answers are the same. There is no difference in the answer. But the condition is that, we reject H naught if t-square. This is the statistic which is greater than the table value or the critical value. So, we reject H naught.