91Ó°ÊÓ

In the \(2010-2011\) season, the Dallas Mavericks won the NBA championship. The two-way table below displays the relationship between the outcome of each game in the regular season and whether the Mavericks scored at least 100 points. $$ \begin{array}{lccc} \hline & 100 \text { or more points } & \text { Fewer than } 100 \text { points } & \text { Total } \\ \text { Win } & 43 & 14 & 57 \\ \text { Loss } & 4 & 21 & 25 \\ \text { Total } & 47 & 35 & 82 \\ \hline \end{array} $$ Which of the following is the best evidence that there is an association between the outcome of a game and whether or not the Mavericks scored at least 100 points? (a) The Mavericks won 57 games and lost only 25 games. (b) The Mavericks scored at least 100 points in 47 games and fewer than 100 points in only 35 games. (c) The Maverick won 43 games when scoring at least 100 points and only 14 games when scoring fewer than 100 points. (d) The Mavericks won a higher proportion of games when scoring at least 100 points \((43 / 47)\) than when they scored fewer than 100 points \((14 / 35)\) (e) The combination of scoring 100 or more points and winning the game occurred more often (43 times) than any other combination of outcomes.

Step by step solution

01

Understand the Problem

The problem given is a two-way table showing the outcome of games for the Dallas Mavericks based on their scoring in the 2010-2011 NBA season. We need to identify the option that best shows an association between the game outcome and whether they scored at least 100 points.

02

Analyze Each Option

We need to consider each option: - **Option (a):** Talks about more wins than losses, but does not specifically connect scoring to these outcomes. - **Option (b):** Mentions the number of games scored above or below 100 points but lacks details on outcome association. - **Option (c):** Compares wins while scoring above or below 100 points, hinting at an association. - **Option (d):** Directly compares the proportion of wins when scoring 100 or more points versus fewer than 100 points, indicating an association with winning. - **Option (e):** States frequencies of particular outcomes without proportion comparison.

03

Calculating Proportions

To compare proportions:- When scoring 100 or more points, the Mavericks had winning outcomes in \( \frac{43}{47} \approx 0.915 \) or 91.5% of the times.- When scoring fewer than 100 points, they had winning outcomes in \( \frac{14}{35} \approx 0.4 \) or 40% of the times.

04

Determine Best Evidence of Association

The best evidence of association between game outcome and scoring at least 100 points is how often scoring above 100 leads to a win when compared to scoring below 100. The option that reflects this comparison with proportions is (d).

05

Conclusion

Given the analysis, option (d) correctly illustrates that the Mavericks were more likely to win when they scored at least 100 points, showing a clear association between points scored and winning outcomes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Statistical Association

When we talk about statistical association in this context, we are exploring whether there is a relationship between two variables: the game outcomes for the Dallas Mavericks and their scoring. Essentially, we're looking to see if winning or losing a game is associated with the number of points scored—either 100 or more, or fewer than 100.
Observing a statistical association helps us determine if one event (scoring a certain number of points) tends to happen with another event (winning the game). It's key to note that association does not imply causation. Just because there is an association, it does not mean scoring fewer points "causes" losses; it is merely correlated.
The two-way table presented lists these occurrences, and our task is to decipher the degree of relationship between the game outcomes and scoring.

Delving into Proportion Comparison

Proportion comparison is a powerful tool in data analysis, especially in determining statistical associations. It allows us to quantify how often an event is likely under different conditions. In our scenario, we compare the proportion of Maverick wins when they score at least 100 points against when they score fewer than 100 points.
Calculating proportions is straightforward:

• When scoring 100 or more, the Mavericks win 43 times out of 47 games. This proportion is calculated as \( \frac{43}{47} \), approximately 91.5%.
• In contrast, when scoring fewer than 100, they win 14 out of 35 games, or \( \frac{14}{35} \), which is about 40%.

Proportion comparison shows a significant difference in winning likelihood depending on points scored, suggesting a potential association between high scoring and game victories.

Game Outcome Analysis: Evaluating Performance

Game outcome analysis involves examining the results of each basketball game and understanding the patterns that emerge based on different scenarios. For the Mavericks' 2010-2011 season, analyzing the outcome of games through the two-way table helps us see patterns like trends in winning based on scoring strategy.
In the provided data, we notice that scoring more than 100 points frequently aligns with a higher likelihood of winning. Specifically, this occurs 43 times compared to the 14 wins when they scored under 100 points. This suggests strategy adjustments could potentially be considered for optimizing performance.
Evaluating performance highlights the importance of setting and achieving certain thresholds—like scoring over 100 points—as they significantly impact the positive game outcomes and the team’s overall strategic approach.