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Let’s go through each question based on the provided scenario.

### 1. Explain what it means when fans claim that Thompson had the hot hand.

When fans claim that Klay Thompson had the "hot hand," they mean that they believe he entered a streak where his probability of making a shot increased following consecutive successful shots. In essence, fans perceive that Thompson’s recent success influences his current performance, making him more likely to succeed again. This phenomenon is often seen as a player's performance being "on fire" or being "in the zone," where each successful attempt seems to improve the likelihood of subsequent successes. However, statistically, the "hot hand" theory suggests a deviation from random independence, implying that a player's chances of success are higher due to prior successful attempts.

### 2. If we wanted to investigate if Thompson's **ability** to make a shot is greater following a made shot than following a missed shot, what hypotheses should we test?

To test if Thompson's ability to make a shot is influenced by his previous shot, we would set up the following hypotheses:

- **Null Hypothesis (H₀):** Thompson's probability of making a shot is independent of his previous shot's outcome. In other words, the probability of making a shot following a made shot is equal to the probability of making a shot following a missed shot.
  
- **Alternative Hypothesis (H₁):** Thompson's probability of making a shot is greater following a made shot than following a missed shot. This would imply that there is a "hot hand" effect, where a successful shot increases the likelihood of subsequent successful shots.

These hypotheses would allow us to analyze whether there is a statistically significant difference in shooting success based on the outcome of the previous attempt.

### 3. Describe a Type I and a Type II error in the context of this problem.

In the context of this problem, where we are investigating whether Thompson has a greater likelihood of making a shot after a successful attempt (hot hand effect):

- **Type I Error (False Positive):** This occurs if we incorrectly reject the null hypothesis. In this case, a Type I error would mean concluding that Thompson does have a hot hand effect (increased probability of making a shot following a successful shot) when, in reality, his shots are independent, and there is no hot hand effect. This error implies observing a pattern in shot success that does not truly exist.

- **Type II Error (False Negative):** This occurs if we fail to reject the null hypothesis when it is actually false. In this context, a Type II error would mean concluding that Thompson's shot outcomes are independent (no hot hand effect) when, in reality, his probability of making a shot does increase following a successful attempt. This error would mean missing a genuine hot hand effect if it exists.

Would you like further clarification on any of these points? Here are some additional questions to deepen understanding:

1. How could we statistically measure the probability of Thompson's shot success following made and missed shots?
2. What statistical test would be appropriate to analyze the dependency of Thompson's shots on previous attempts?
3. Why might fans perceive a "hot hand" even if statistically the shots are independent?
4. Can the "hot hand" phenomenon be applied to other sports or activities, and how might we test it?
5. How could the concepts of Type I and Type II errors influence sports analysts’ predictions?

**Tip:** When testing hypotheses, always set a significance level (e.g., 0.05) to control the probability of making a Type I error.

On January 23, 2015, basketball player Klay Thompson of the Golden State Warriors scored 52 points in a victory over the Sacramento Kings, including 37 in the 3rd quarter alone. During the game, many fans suggested that Thompson had the hot hand. However, even though this was a truly amazing performance, it is possible that the outcomes of Thompson’s shots were independent, and he didn’t have the hot hand. Overall, he made 16 shots in 25 attempts. 1. Explain what it means when fans claim that Thompson had the hot hand. 2. If we wanted to investigate if Thompson’s ability to make a shot is greater following a made shot than following a missed shot, what hypotheses should we test? 3. Describe a Type I and a Type II error in the context of this problem.

Explore the concept of the hot hand phenomenon using Klay Thompson’s 52-point game. Learn how hypothesis testing and probability help assess whether his shot-making ability improved after successful attempts, along with the implications of Type I and Type II errors. Suitable for advanced high school students.

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