## Addendum on Pythagorean Expectation May 20, 2010

Posted by tomflesher in Baseball, Economics.
Tags: , , ,
Making the standard statistical assumptions, the margin of error using proportions is $\sqrt{\frac{p(1-p)}{n}}$ . Three of the proportions were .46, meaning that the margin of error would be $\sqrt{\frac{.46(.54)}{13}} = \sqrt{\frac{.2484}{13}}$ which simplifies to $\sqrt{.0191} = {.1382}$. Using 12 degrees of freedom, a t-table shows that the critical value for 95% confidence  is 2.18. Thus, the binomial confidence interval method, tells us we can be 95% sure that the true value of the proportion lies within the range .46 ± 2.18*.1382 = .46 ± .30 = .16 … .76. Clearly, this range is far too large to reject the conclusion that the proportion is significantly different from .5.
For the simple measure of more runs, the proportion was .31, meaning that the margin of error is $\sqrt{\frac{.31(.69)}{13}} = \sqrt{\frac{.2139}{13}}$ or $\sqrt{.0165} = {.1283}$. The 95% confidence interval around .31 is .31 ± 2.18*.1283 = .31 ± .2797 = .03 … .59. Again, .5 is included in this range.