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The image you uploaded describes a situation involving goalkeepers deciding between "diving" or "staying" in place during penalty kicks, using the context of "action bias." Here's a breakdown of how to proceed with the tasks in the image:

### 1. Null and Alternative Hypotheses

- **Null Hypothesis (H₀):** There is no significant difference in the success rates (save percentages) between staying in the center and diving. In other words, any observed difference in performance is due to random chance.

- **Alternative Hypothesis (Hₐ):** There is a significant difference in the success rates between staying in the center and diving. Specifically, one strategy may be more effective than the other (diving vs staying).

### 2. Test Statistic Calculation
To calculate the test statistic, you would compare the proportion of successful saves when the goalkeeper "stayed" versus when they "dived."

- From the data provided:
  - For "Stay": \( 6/18 \) = 0.333 (33.3%)
  - For "Dive": \( 232/268 \) = 0.865 (86.5%)
  
  The difference in proportions is:
  \[
  \text{Difference} = 0.865 - 0.333 = 0.532
  \]
  
  Next, calculate a **z-test** or **chi-square test** for the difference in proportions, depending on how you're asked to proceed.

### 3. Simulation (Index Card Method)
For the index card simulation:

- You would randomly assign outcomes (whether the goalkeeper stayed or dived) to cards. Each card represents either a goal or a save.
- Shuffle the cards and simulate the decision-making process of the goalkeepers repeatedly (e.g., 100 or 1000 times).
- Each time, compute the difference in the number of saves between "staying" and "diving" and plot these differences to form a dotplot.
- This simulation allows you to estimate whether the observed difference (0.532) is likely to have occurred by chance.

### 4. Comparing Simulation to Performance
- Finally, compare the actual performance difference (0.532) with the results from your simulation.
- If the difference observed in the simulation under random chance rarely exceeds 0.532, you would reject the null hypothesis and conclude that the difference is statistically significant, suggesting that diving or staying is indeed a better strategy.

Do you need more detailed steps for the calculations or simulation process?

### Follow-up Questions:
1. Would you like a detailed explanation of how to calculate the **test statistic** (z-test or chi-square)?
2. Do you need help setting up the **simulation** using a specific tool?
3. Are there any other data points or variables in the problem that need clarification?
4. Do you want to know how to interpret the **dotplot** in more detail?
5. Should I elaborate on how to present the final **conclusion** based on the results?

#### Tip:
When performing hypothesis testing, always check the assumptions of your test, like sample size and normality, to ensure the results are valid.

The desire to 'do something' is very strong for goalkeepers in soccer when facing a penalty kick. Using the data provided, analyze if staying in the center or diving is a more effective strategy.

Analyze whether staying in the center or diving is a more effective strategy for goalkeepers facing penalty kicks. This problem involves hypothesis testing, proportions, and simulation techniques using real soccer data.

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