Jason usually makes 30% of his 3-point attempts, but in the last 2 games, he tried 16 3-point shots and made 8 of them. Is he getting better, or is it possible that this difference could reasonably happen by random chance? Use the stapple.com to run a simulation with 100 performances. How many attempts is it out of

To determine if Jason's recent shooting performance is an indication of improvement or if it could reasonably happen by random chance, we can run a simulation. However, since I don't have access to external websites like stapple.com, I can guide you through the steps to simulate this using basic statistical concepts.

### Steps to Simulate:
1. **Jason's expected performance**: Normally, Jason makes 30% of his 3-point shots. So, in any set of attempts, the probability of success (making the shot) is \( p = 0.3 \).
2. **His recent performance**: In the last 2 games, he made 8 out of 16 attempts, which is 50% (a better performance compared to his usual rate).
3. **Simulation setup**: We will simulate 100 performances where Jason attempts 16 shots with a success probability of 30%, and we'll count how often he achieves 8 or more successful shots (which corresponds to the performance in question).

### Statistical Analysis:
- **Null Hypothesis**: Jason's shooting ability has not improved; the difference is due to random chance.
- **Alternative Hypothesis**: Jason's shooting ability has improved, and the difference is not just random variation.
  
This will involve running a **binomial simulation** to see how many times, out of 100 trials, Jason can hit 8 or more shots out of 16 if the probability of success remains 30%.

### Running the Simulation Manually (If stapple.com is not available):
1. The number of shots attempted is \( n = 16 \).
2. The probability of making a shot is \( p = 0.3 \).

You can simulate this using software such as Excel, Python, or R. Here's a Python-like explanation to run it:
```python
import numpy as np

# Parameters
p = 0.3  # Probability of making a 3-point shot
n = 16   # Number of attempts per game
trials = 100  # Number

Jason usually makes 30% of his 3-point attempts, but in the last 2 games, he tried 16 3-point shots and made 8 of them. Is he getting better, or is it possible that this difference could reasonably happen by random chance?

Explore whether Jason's improved 3-point shooting performance is due to random chance or genuine improvement. This solution uses binomial probability and hypothesis testing to simulate Jason's 16 shot attempts, with a 30% success rate. Suitable for students studying probability or statistics.

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