## Saturday, March 1, 2014

### Understanding Confidence Intervals with MLB win percentage data.

Question:   The table below contains data on the winning percentage for MLB teams in 1996.   Construct a 90% confidence interval for the winning percentage for each of these teams.

For what teams does this confidence interval suggest that that the true probability of winning a game differs from 0.50?

Based on a 90% confidence interval how many teams can claim that their win probability might be 60%?

 Winning Percentage MLB Teams in 1996 p Atlanta 0.593 Chicago 0.469 Cincinnati 0.500 Colorado 0.512 Florida 0.494 Houston 0.506 Los Angeles 0.556 Montreal 0.543 New York 0.438 Philadelphia 0.414 Pittsburgh 0.451 San Diego 0.562 San Francisco 0.420 Saint Louis 0.543 Baltimore 0.543 Boston 0.525 California 0.435 Chicago 0.525 Cleveland 0.615 Detroit 0.327 Kansas City 0.466 Milwaukee 0.494 Minnesota 0.481 New York 0.568 Oakland 0.481 Seattle 0.528 Texas 0.556 Toronto 0.457

Answer:  The lower bound of a 90% confidence interval is p – 1.65 x SE.    The upper bound of the 90% confidence interval is p +1.65 x SE.

The standard error SE=((p x (1-p))/n)0.5 where n=162 is the number of games played.

The table below contains the win proportion, the standard error, and the lower and upper bound of confidence intervals for the winning proportion

 Confidence Interval for Winning Percentages for MLB Teams in 1996 p SE Lower Bound Upper Bound Atlanta 0.593 0.039 0.529 0.657 Chicago 0.469 0.039 0.404 0.534 Cincinnati 0.500 0.039 0.435 0.565 Colorado 0.512 0.039 0.447 0.577 Florida 0.494 0.039 0.429 0.559 Houston 0.506 0.039 0.441 0.571 Los Angeles 0.556 0.039 0.492 0.620 Montreal 0.543 0.039 0.478 0.608 New York 0.438 0.039 0.374 0.502 Philadelphia 0.414 0.039 0.350 0.478 Pittsburgh 0.451 0.039 0.386 0.516 San Diego 0.562 0.039 0.498 0.626 San Francisco 0.420 0.039 0.356 0.484 Saint Louis 0.543 0.039 0.478 0.608 Baltimore 0.543 0.039 0.478 0.608 Boston 0.525 0.039 0.460 0.590 California 0.435 0.039 0.371 0.499 Chicago 0.525 0.039 0.460 0.590 Cleveland 0.615 0.038 0.552 0.678 Detroit 0.327 0.037 0.266 0.388 Kansas City 0.466 0.039 0.401 0.531 Milwaukee 0.494 0.039 0.429 0.559 Minnesota 0.481 0.039 0.416 0.546 New York 0.568 0.039 0.504 0.632 Oakland 0.481 0.039 0.416 0.546 Seattle 0.528 0.039 0.463 0.593 Texas 0.556 0.039 0.492 0.620 Toronto 0.457 0.039 0.392 0.522

Only three major league teams -- the Atlanta Braves, the Cleveland Indians and the New York Yankees -- had a lower bound for this confidence interval that exceeded 0.500.  Based on a 90% confidence interval analysts can be confident that their win probability was greater than 0.50.

Four teams -- Philadelphia, San Francisco, California and Detroit  -- all had an upper bound below 0.500. Based on a 90% confidence interval analysts can be confident that the win probability of these four teams is less than 0.50.

Nine teams -- Atlanta, Los Angeles, Montreal, San Diego, Saint Louis, Baltimore, Cleveland, New York and Texas  had confidence interval with an upper bound over 0.600 and a lower bound less than 0.60.  Analysts cannot rule out a win probability of 0.60 based on this confidence interval.