- Calculate each team's win ratio: Eagles \(\approx 0.4583\), Dolphins \(\approx 0.5238\), Bengals \(= 0.3125\), Hawks \(= 0.36\), Baboons \(\approx 0.2581\), Elephants \(\approx 0.2333\), Zebras \(\approx 0.2222\).
- Part (a): Only Dolphins satisfy \(R > 0.5\), so the team is \(\boxed{\text{Dolphins}}\).
- Part (b): Comparing differences for \(R < 0.5\), the Eagles have the smallest difference (\(0.0417\)), so the team is \(\boxed{\text{Eagles}}\).
- Part (c): Comparing differences for \(R > 0.3333\), the Hawks have the smallest difference (\(0.0267\)), so the team is \(\boxed{\text{Hawks}}\).
Explanation
- Calculating Win Ratios
To solve the problem, we calculate the win ratio for each team by dividing the number of Games Won (GW) by the number of Games Played (GP). The data for the seven teams is given as:
- Eagles: \(\frac{11}{24} \approx 0.4583\)
- Dolphins: \(\frac{11}{21} \approx 0.5238\)
- Bengals: \(\frac{10}{32} = 0.3125\)
- Hawks: \(\frac{9}{25} = 0.36\)
- Baboons: \(\frac{8}{31} \approx 0.2581\)
- Elephants: \(\frac{7}{30} \approx 0.2333\)
- Zebras: \(\frac{6}{27} \approx 0.2222\)
We use these calculated ratios to determine which teams best fit the criteria for parts (a), (b), and (c).
- Analysis for Part (a)
For part (a), we seek the team with a win ratio \(R > \frac{1}{2} = 0.5\) and the smallest positive difference \(R - 0.5\). Comparing the ratios, only the Dolphins have a win ratio greater than \(0.5\) (specifically, \(\frac{11}{21} \approx 0.5238\)). Therefore, the Dolphins are the only team that satisfies the condition for (a). The difference is \(0.5238 - 0.5 = 0.0238\). Thus, the team is the Dolphins.
- Analysis for Part (b)
For part (b), we look for the team with \(R < 0.5\) and the smallest positive difference \(0.5 - R\). The teams with \(R < 0.5\) and their differences are:
- Eagles: \(0.5 - 0.4583 = 0.0417\)
- Bengals: \(0.5 - 0.3125 = 0.1875\)
- Hawks: \(0.5 - 0.36 = 0.14\)
- Baboons: \(0.5 - 0.2581 = 0.2419\)
- Elephants: \(0.5 - 0.2333 = 0.2667\)
- Zebras: \(0.5 - 0.2222 = 0.2778\)
The smallest difference is \(0.0417\), corresponding to the Eagles.
- Analysis for Part (c)
For part (c), we look for the team with \(R > \frac{1}{3} \approx 0.3333\) and the smallest positive difference \(R - \frac{1}{3}\). The teams with \(R > 0.3333\) are the Eagles (\(0.4583\)), the Dolphins (\(0.5238\)), and the Hawks (\(0.36\)). Their differences from \(\frac{1}{3}\) are:
- Eagles: \(0.4583 - 0.3333 = 0.125\)
- Dolphins: \(0.5238 - 0.3333 = 0.1905\)
- Hawks: \(0.36 - 0.3333 = 0.0267\)
The smallest difference is \(0.0267\), corresponding to the Hawks. Thus, the Hawks satisfy the condition for (c).
Examples
In professional sports, win ratios are frequently analyzed to evaluate team performance against specific benchmarks, such as winning at least half of their games. By calculating these ratios and comparing them to target thresholds (like 50% or 33%), team managers can identify which teams are performing just above or below a desired level of competitiveness. This helps in strategic planning, identifying areas for improvement, and assessing relative team strength in a league standings table.