Wednesday, February 3, 2016

Home Field Advantage in Three Sports


Home Field Advantage in Three Sports

This post attempts to assess the importance of the home field in three sports – professional basketball, football and baseball.   The analysis in this post involves an evaluation of all games played by all teams in the league. The data for baseball and football cover the 2015 regular season.   The data for basketball covers the 2014 to 2015 regular season.

Data:  The table below contains information on games won by the home team and games won by the road team in three professional sports.

Home Wins and Losses Across Sports
Games Won by Home Team
Games Won By Away Team
Baseball
1315
1114
Football
138
118
Basketball
707
523


Questions:

·      What percent of all games played did the home team win in each sport?

·      Create a 95% confidence interval for the home-win proportion in each sport?

·      Explain why it is unnecessary to conduct an evaluation of the home-loss percentage in each sport if one has already conducted an analysis of the home-win percentage?

·      Test the hypothesis that the home-win proportion for each team is greater than 0.5 in each sport

·      Explain how the number of games played impacts the estimated standard error used in the t-test for the hypothesis that the home-win proportion differs from O.5.



Analysis:


What percent of all games played did the home team win in each sport?  Discuss.

The point estimates suggest the home-field advantage is highest in basketball.  This may occur because the fans are so close to the court.   However, courts in basketball and football are identical while baseball stadiums differ in dimensions.   The differences in stadiums could allow an organization to buy players who perform well in certain fields.   For example, a team with a short left field would load up on right-handed power hitters.


Home-Win Proportions in Three Sports
Games Won by Home Team
Games Won By Away Team
Total
Home win Proportion
Baseball
1315
1114
2429
0.5414
Football
138
118
256
0.5391
Basketball
707
523
1230
0.5748



Create a 95% confidence interval for the home-win proportion in each sport?


The z-score used to construct the 95% confidence interval is 1.96.   The upper and lower bounds for the confidence interval are the obtained by adding/subtracting 1.96 x SE.  The SE is  (px(1-p)/n)0.5 where p is the estimated home-win proportion and n is the sample size.


Calculation of 95% Confidence of Home-Win Proportion in Three Sports
Sport
Estimator of Home Win Proportion
Sample Size
Standard Error
Lower Bound of CI
Upper Bound of CI
Baseball
0.5414
2429
0.0101
0.5216
0.5612
Football
0.5391
256
0.0312
0.4780
0.6001
Basketball
0.5748
1230
0.0141
0.5472
0.6024



Explain why it is unnecessary to conduct an evaluation of the home-loss percentage in each sport if one has already conducted an analysis of the home-win percentage?


The event win at home and the event lose at home are complements.  If a team wins 60% of home games it loses 40% of home games.   If 0.5 is outside the confidence of the home win proportion it is also outside the confidence of the home loss proportion.  

Note also the standard errors of the confidence interval for the home win proportion is identical to the standard error of the home-loss proportion.  


Test the hypothesis that the home-win percentage for each team is greater than 0.5 in each sport.  Use a one-tailed test at a 0.025 level of significance.  Explain how the number of games played impacts the estimated standard error used in the t-test for the hypothesis that the home-win proportion differs from O.5.


The calculations for these tests are laid out in the table below.


Test results for the hypothesis that home win proportion
is greater than 0.05.
Sport
Estimator of Home Win Proportion
Value Of Proportion Under the Null Hypothesis
Sample Size
Standard Error
T-Statistic
Critical Value for 0.01 level of significance
Baseball
0.5414
0.5
2429
0.0101
4.1
2.33
Football
0.5391
0.5
256
0.0313
1.3
2.33
Basketball
0.5748
0.5
1230
0.0143
5.2
2.33

Discussion:

·      The standard error in the t-statistic is ((0.5 x (1-0.5)/n)^0.5.  Differences in the value of the standard error depend solely on the sample size n since the assumed null hypothesis probability of 0.5 is used in the construction of all three standard errors.  The standard error is a lot larger for football because this sport has fewer games in one season.

·      The critical value for the one tailed test at a 0.0 level of significance can easily be obtained in Excel by inserting NORMSINV(0.99).

·      The null hypothesis is rejected for baseball and basketball but not for football.  The point estimate for the home field win proportion is fairly similar for football and baseball but the home-win proportion is highly significant for baseball and insignificant for football.   This result was determined by the sample size. 

There is of course a small but significant relationship between home field and game outcome in football.   A test for football based on pooled data over multiple seasons will find a significant result.   Sample size often matters a lot both for these sport-math problems and for high dollar questions like the efficacy of different health care treatments. 

Concluding Thoughts:  Often when samples are small proportions are tested with a chi-squared test rather than a t-test.   It is possible to test the hypothesis that the football and baseball home field win percentages are identical by pooling data and using the chi-square test.   I am reluctant to pool the football and baseball data from a single season because the sample size for baseball is so much larger than the sample for football and the pooled sample will be dominated by the larger sample.

People who are interested in comparing the chi-squared and t-tests should consider the following post.


I am working on a new book on statistics and sports.   People who are interested in my approach to the world can look at my work on Kindle and on Teachers Pay Teachers.

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