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.



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