Question

the wakefield high school football team won the regional championship in 2022. a record of their wins and losses is shown, in which the relationship between wins and losses is sorted by number of points scored.

	win	loss	total
< 21 points
total	45		50

does the data give evidence of an association between scoring at least 21 points and the football team winning the game?

there is a weak, negative association.
there is a weak, positive association.
there is a strong, negative association.
there is a strong, positive association.

Explanation:

Step1: Complete the contingency table

First, find the number of wins with <21 points: Total wins are 45, so 45 - 20 = 25.
Number of losses with ≥21 points: Let total losses be L. Total games: 50, so L = 50 - 45 = 5. Then losses with ≥21 points: 5 - 2 = 3.
Number of games with <21 points: Total <21 points games = 25 (wins) + 2 (losses) = 27.
Number of games with ≥21 points: 20 (wins) + 3 (losses) = 23.

The table becomes:
| | Win | Loss | Total |
|----------|-----|------|-------|
| ≥21 pts | 20 | 3 | 23 |
| <21 pts | 25 | 2 | 27 |
| Total | 45 | 5 | 50 |

Step2: Calculate conditional probabilities

Probability of win given ≥21 pts: \( \frac{20}{23} \approx 0.87 \).
Probability of win given <21 pts: \( \frac{25}{27} \approx 0.93 \). Wait, no—wait, actually, we check association direction. Wait, higher points (≥21) have win rate ~0.87, lower points (<21) have win rate ~0.93? Wait, that can’t be. Wait, no, maybe I miscalculated losses. Wait, total losses: 50 - 45 = 5. Losses with <21 points: 2, so losses with ≥21 points: 5 - 2 = 3. Wins with ≥21: 20, so wins with <21: 45 - 20 = 25. So total ≥21: 20 + 3 = 23, <21: 25 + 2 = 27.

Now, probability of win when ≥21: 20/23 ≈ 0.87, when <21: 25/27 ≈ 0.93. Wait, that’s a slight decrease, but maybe I messed up. Wait, no—wait, the question is about association between scoring ≥21 and winning. Wait, maybe the initial table was misread. Wait, original table: Win row, ≥21 is 20, Loss row, ≥21? Wait, no, original table: first column ≥21, Win is 20, Loss is 2? Wait, no, the table is:

Columns: Win, Loss, Total
Rows: ≥21, <21, Total

So original table:
Row ≥21: Win=20, Loss=?, Total=?
Row <21: Win=?, Loss=?, Total=?
Total: Win=45, Loss=?, Total=50

So Loss total: 50 - 45 = 5. So Loss column: 2 (in <21 row) and? in ≥21 row. So ≥21 Loss: 5 - 2 = 3. Win in <21: 45 - 20 = 25. So table:

| | Win | Loss | Total |
|----------|-----|------|-------|
| ≥21 pts | 20 | 3 | 23 |
| <21 pts | 25 | 2 | 27 |
| Total | 45 | 5 | 50 |

Now, P(Win | ≥21) = 20/23 ≈ 0.87, P(Win | <21) = 25/27 ≈ 0.93. Wait, that’s a negative association? But 0.87 < 0.93, so as points increase (≥21), win probability slightly decreases? But that seems weak. Wait, maybe the original table had Loss in ≥21 as 2? Wait, maybe I misread the table. Let me re-examine the image: The table has "Win 20" in ≥21, "Loss 2" in... maybe Loss row? Wait, maybe the table is:

Row: ≥21 points, columns Win=20, Loss=?, Total=?
Row: <21 points, columns Win=?, Loss=?, Total=?
Total: Win=45, Loss=?, Total=50

Wait, maybe the Loss column has 2 in <21, and total Loss is 5, so ≥21 Loss is 3. Win in <21 is 45 - 20 = 25. So the table is correct. Now, the difference in win rates: 20/23 ≈ 0.87 vs 25/27 ≈ 0.93. The difference is small (0.93 - 0.87 = 0.06), so weak. But direction: since higher points (≥21) have lower win rate, that’s a negative association? Wait, no—wait, maybe I got the rows reversed. Wait, maybe the first row is <21, second is ≥21? Wait, the problem says "sorted by number of points scored"—maybe ≥21 is first. Wait, no, the key is to check the association. Let's calculate the odds ratio.

Odds of win for ≥21: 20/3 ≈ 6.67
Odds of win for <21: 25/2 = 12.5
Odds ratio: 6.67 / 12.5 ≈ 0.53, which is less than 1, indicating a negative association (since higher points (≥21) have lower odds of winning than <21). But the difference in probabilities is small (0.87 vs 0.93), so weak negative association? Wait, but the options: "There is a weak, negative association." Wait, but let's check again. Wait, maybe I made a mistake in the table. Wait, total wins: 45, so Win column: 20 (≥21) + 25 (<21) = 45. Total losses: 3 (≥21) + 2 (<21) = 5. Total games: 23 + 27 = 50. Correct.

Now, the association: when points are ≥21, win rate is 20/23 ≈ 87%, when <21, win rate is 25/27 ≈ 93%. So as points increase (from <21 to ≥21), win rate decreases slightly. So negative association, and since the difference is small, weak. So the correct option is "There is a weak, negative association."

Answer:

There is a weak, negative association.