Equality symmetric methods, factorizing

Suppose a factor X has 45, 50, and 55 as extreme low, nominal, and extreme high levels, respectively, and an effect of 100 on response Y, with the critical effect equal to 80. Then the non-significance interval limits for this factor are [46.0,54.0], which means that when restricting the levels of X to this interval, the quantitative aspect of the method is considered robust. It can be noticed that the interval is symmetrically around the nominal level and meant for factors thus examined, i.e., with extreme levels symmetrically around the nominal. 

Formation of mixed anhydrides poses an even greater problem using this technique. If propanoic acid (17) and butanoic acid (7) react in the presence of an acid catalyst, propanoic acid may react with HCl to form 56, but butanoic acid can also react to form 12. In the absence of some unknown factor, there is an equal chance of forming either cation. If 56 reacts with another molecule of propanoic acid, the symmetrical 55 is formed, but if 56 reacts with a molecule of butanoic acid, 57 is formed. Similarly, cation 12 may react with propanoic acid to form 57, but it can also react with another molecule of butanoic acid to form 58. Forming either 55 or 58 has one chance and forming 57 has two chances in fact, however, all three anhydrides are generated by this reaction in a ratio of 1 2 1 (55 57 58). This is said to be a statistical mixture of products. The use of acid chlorides to form mixed anhydrides is a superior method to this statistical coupling of two acids. 

The method of reduction is clear from the transition between the first and the second determinant. We multiply the elements of the second row with the appropriate number and subtract them from the elements of the other rows, in such a way that the elements of the second column (except for the diagonal element) become equal to zero. The remaining elements of the second row may then be neglected. Then the determinant is again symmetric. The final result is a simple product, of which the latter factor is almost zero, namely —0.00005. However, if we put = -0.15200, we find with a similar calculation that... 

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