Experimental tests are to be performed to determine whether a new catalyst A is superior to the current catalyst B, based on their yields for a chemical reaction. Denote the yields by XA and XB and their standard deviations by σA = 3% and σB = 2%. What is the standard deviation for the difference i

Question

Experimental tests are to be performed to determine whether a new catalyst [latex]A[/latex] is superior to the current catalyst [latex]B[/latex], based on their yields for a chemical reaction. Denote the yields by[latex]X_{A}[/latex] and [latex]X_{B}[/latex] and their standard deviations by [latex]\sigma _{A}[/latex] = 3% and [latex]\sigma _{B}[/latex] = 2%. What is the standard deviation for the difference in yields, [latex]X_{A} − X_{B}[/latex]?

Accepted Answer

Let [latex]Y=X_{A} +X_{B}[/latex], an expression in the form of Eq. F-13 with [latex]c_{A} = 1[/latex] and [latex]c_{B} = −1[/latex]. Thus Eq. F-15 becomes

[latex]Y=\sum\limits_{i=1}^{P}{c_{i}X_{i} }[/latex] (F-13)

[latex]\sigma ^{2}_{Y} = \sum\limits_{i=1}^{P}{c^{2}_{i}\sigma ^{2}_{i} }[/latex] (F-15)

[latex] \sigma ^{2}_{Y}=\sigma ^{2}_{A}+\sigma ^{2}_{B}[/latex]

and

[latex] \sigma _{Y} = \sqrt{\sigma ^{2}_{A}+\sigma ^{2}_{B}}= \sqrt{(3\%)^2+(2\%)^2} = 3.6\%[/latex]

Thus, the standard deviation of the difference is larger than the individual standard deviations.

Question F.4: Experimental tests are to be performed to determine whether ...

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