The example about averages of bowling scores at the beginning of this section is an instance of what is known as Simpson's paradox. The following example illustrates this paradox.

Notice that Player A has a better overall batting average than Player B, but yet is worse against both right-handed pitchers and left-handed pitchers. The following is an algebraic statement of Simpson's paradox.
Consider two populations for which the overall rate \(r\) of occurrence of some phenomenon in population A is greater than the corresponding rate \(R\) in population B. Suppose that each of the two populations is composed of the same two categories \(C_{1}\) and \(C_{2}\), and the rates of occurrence of the phenomenon for the two categories in population A are \(r_{1}\) and \(r_{2}\), and in population B are \(R_{1}\) and \(R_{2}\). If \(r_{1}R\), then Simpson's paradox is said to have occurred.

a. Relate the variables in this statement to the numbers in the example.

b. An example of Simpson's paradox from real life is the fact that the overall federal income tax rate increased from 1974 to 1978, but decreased for each bracket. Make up a fictitious example to show how this might be possible.

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Against right-handed pitchers Against left-handed pitchers Overall At Bat 202 250 452 Player A Hits 45 71 116 Average 0.223 0.284 0.257