 This is example number 6, the hotelling T-Skill for two samples, different examples, different methods. We have done one sample, two samples, this is just for the practice. Now the example is, 50 bars of soap are manufactured in such two ways. Now the two characteristics X1 and the second characteristic is the X2. The summary statistic for bars produced by method 1 and the method 2. This is the X bar 1 and S1, X bar 2, this is the soap 1 characteristic and this is the soap 2 characteristic. Here is the sample S1 and this is the S2. Now test the hypothesis, there are two samples. We have basically measured the characteristics of two samples, X1 and S2. These are the characteristics of the soap. Summary statistics, we have given its mean, variances, and then test the hypothesis at 5% or two, we have to find the tail test. So we have total 50 bars of soap and we are measuring its two different characteristics. This is the question and the solution, null and alternative hypothesis, this is two sided, level of significance, 5%, this is the test statistic. Now what is the difference in the test statistic? Here is SP, pooled standard deviation or pooled variance we are taking here, this is the SP. Given is given and here pooled we have to take variances. Now how will we find SP? Now this is the calculation X bar 1 minus X bar 2 is given. Now here is the SP, pooled standard deviation which is equal to 1 over N1 plus N2 minus 2 and this is the formula for finding the value of SP. Now SP, pooled standard deviation or pooled variance we are taking because variance has a covariance matrix, so we are saying pooled variance. 1 over N1, take 50 bars N2 because we have 50 bars, we have measured two characteristics, P which is equal to 2, so N1 minus 1, S1, N2 minus 2, S2. After calculating SP, we have the final result of 2 cross 2 matrix, this is equal to SP. Now for the value of T square, we have to put values in the statistic. Basically, the difference here is we have used the pooled standard deviation or pooled variance concept. Further, after solving this formula, the final, the T square which is equal to 53.546, we have the value of the test statistic, this is the calculated value. Now for critical region, for two samples which is equal to this one, here alpha equals to 0.05 and alpha by 2 which is equal to 0.025. P2, you get total of my past two dimension P which is equal to 2. Now 297, alpha by 20.025, 297, 2 and 97. Again, you know that there is no value on 97, here we have 120 by 3.8. Further, here we have the value with which we are multiplying which is equal to 3.8046. After multiplying, the final result is 7.685 and the calculated value is the 53.546 and the table value is 7.685. The calculated value is greater than the critical value, so we reject H0 at 5% level of significance. So, what did we do in this? Further, we checked the 50 bars, their two characteristics and further we have the answer. So, we reject H0 at 5% level of significance. This is the another example of the hoteling T square for two samples.