Question 1201173
**a) Hypothesis Test**

* **Hypotheses:**
    * **Null Hypothesis (H0):**  p1 - p2 = 0 (There is no difference in the proportion of people who got sick between those who ate fish and those who did not.)
    * **Alternative Hypothesis (H1):** p1 - p2 > 0 (The proportion of people who ate fish and got sick is significantly higher.)

* **Sample Proportions:**
    * p1_hat = 64/80 = 0.8 (Proportion of those who ate fish and got sick)
    * p2_hat = 39/60 = 0.65 (Proportion of those who did not eat fish and got sick)

* **Pooled Proportion:**
    * p̂ = (64 + 39) / (80 + 60) = 103 / 140 ≈ 0.7357

* **Standard Error:**
    * SE = √[p̂(1-p̂)(1/n1 + 1/n2)] 
        * SE = √[0.7357 * (1 - 0.7357) * (1/80 + 1/60)] 
        * SE ≈ 0.0683

* **Test Statistic (z-score):**
    * z = (p1_hat - p2_hat) / SE 
        * z = (0.8 - 0.65) / 0.0683 
        * z ≈ 2.20

* **P-value:**
    * Using a standard normal distribution table, find the p-value associated with z = 2.20. 
    * P-value ≈ 0.0139

* **Decision:**

    * Since the p-value (0.0139) is less than the significance level (α = 0.05), we **reject the null hypothesis**. 

* **Conclusion:**

    * There is sufficient evidence to suggest that the proportion of people who ate fish and got sick is significantly higher than the proportion of people who did not eat fish and got sick. This provides some evidence that the fish may have contributed to the illness.

**b) Type of Error**

* Since we rejected the null hypothesis, we could have made a **Type I error**.

* **Type I Error:** 
    * This occurs when we reject the null hypothesis when it is actually true. 
    * In this context, a Type I error would mean concluding that the fish caused the illness when in reality, there was no significant difference in the proportion of people who got sick between those who ate fish and those who did not.

* **Interpretation of Type I Error:**

    * This could lead to unnecessary concern and potentially harmful actions, such as unnecessarily restricting the sale or consumption of fish, even if it was not the cause of the illness.

**Important Notes:**

* This analysis provides some evidence of a possible link between eating the fish and illness, but it does not definitively prove causation. 
* Other factors could have contributed to the illness, such as other food items, cross-contamination, or other environmental factors. 
* Further investigation and a more comprehensive analysis would be necessary to establish a definitive cause-and-effect relationship.

This analysis provides a framework for understanding the potential implications of a Type I error in this specific context.