Some people have argued that the Pythagorean Theorem of Baseball is a better predictor of a team’s ability than its winning percentage:

where W is a predictor of what fraction of games a team is expected to win, s is the number of runs scored, and a is the number of runs allowed.

(a) The team with the best record in the 2019 season was the Houston Astros, with 107 wins and 55 losses. They scored 920 runs and allowed 640 runs. Calculate and compare their winning percentage (as a decimal) and their Pythagorean Theorem score W.
(b) Keeping their runs scored at 920 and considering their runs allowed as a variable a, find dW/da when a = 640.
(c) Interpreting your answer to part (b) as the approximate amount that W would change if a increased by one run, use your answer to approximate W when a = 641, and then calculate the actual value.
(d) Some people have argued that the exponent of s and a in the formula for W should be 1.81 rather than 2. Calculate W (with s = 920 and a = 640) and dW/da with this exponent.

Transcribed Image Text:

W = $² s² + a²¹