To determine the best evidence of an association between scoring 40 points or more and the rugby team winning a game, let's analyze the information given in the table in more detail.
1. The Bimini team won 29 games and lost only 11 games:
- This shows the overall success of the team but does not specifically address the relationship between the points scored and the outcome of the game.
2. The Bimini team scored 40 points or more in 20 games and less than 40 points in 20 other games:
- This describes the distribution of the number of games based on scoring but does not reflect the outcomes of these games in terms of wins and losses.
3. The Bimini team won a high proportion of games when scoring 40 points or more [tex]\(\frac{18}{20}\)[/tex] compared with when they scored less than 40 points [tex]\(\frac{11}{20}\)[/tex]:
- Here, we are given specific proportions of wins based on the scoring threshold:
- When scoring 40 points or more, the proportion of wins is [tex]\( \frac{18}{20} \)[/tex] which equals 0.9.
- When scoring less than 40 points, the proportion of wins is [tex]\( \frac{11}{20} \)[/tex] which equals 0.55.
- These proportions clearly demonstrate that the team is more likely to win when they score 40 points or more.
4. The combination of scoring 40 points or more and winning occurred more times than any other outcome:
- Counting the frequencies:
- Winning with [tex]\( \geq 40 \)[/tex] points: 18 times
- Winning with [tex]\( < 40 \)[/tex] points: 11 times
- Losing with [tex]\( \geq 40 \)[/tex] points: 2 times
- Losing with [tex]\( < 40 \)[/tex] points: 9 times
- It is true that winning with [tex]\( \geq 40 \)[/tex] points (18 times) is the most frequent single outcome, but this alone does not provide a comparative analysis to establish a clear relationship between scoring and winning.
Given these points, the best evidence of association comes from the comparison of win proportions based on scoring thresholds:
The Bimini team won a high proportion of games when scoring 40 points or more [tex]\(\frac{18}{20}\)[/tex] compared with when they scored less than 40 points [tex]\(\frac{11}{20}\)[/tex].
This clearly shows a higher win ratio when the team scores 40 points or more, providing strong evidence of an association between higher scoring and winning a game. Thus, the statistical comparison of win proportions is the most compelling evidence of the association.
