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 (1p))/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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