Question 1194131
**1. Define the Variables:**

* **n:** Number of attempts = 250
* **p:** Probability of winning a prize = 0.15
* **X:** Number of prizes won

**2. Calculate the Expected Number of Prizes Won:**

* Expected number of prizes = n * p = 250 * 0.15 = 37.5

**3. Determine the Standard Deviation:**

* Standard deviation (σ) = √(n * p * (1 - p)) = √(250 * 0.15 * 0.85) ≈ 5.63

**4. Use Normal Approximation (Assuming n*p and n*(1-p) are both greater than 5):**

* Since n*p = 37.5 and n*(1-p) = 212.5, both are greater than 5, we can use the normal approximation to the binomial distribution.

* **Calculate the Z-score for 95% Confidence:**
    * For a 95% confidence level, the Z-score is approximately 1.96 (from the standard normal distribution table).

* **Calculate the Upper Bound for the Number of Prizes:**
    * Upper Bound = Expected Value + (Z-score * Standard Deviation)
                   = 37.5 + (1.96 * 5.63) ≈ 48.6

**5. Conclusion:**

* Based on the normal approximation, Yuen can be at least 95% confident that the number of prizes won will be less than or equal to 48.6. 

* Since she has 45 prizes, there is a possibility that she might not have enough prizes for all the winners at the 95% confidence level.

**Important Note:**

* This analysis uses a normal approximation, which may have some slight inaccuracies. For more precise calculations, you could use the binomial distribution directly (using statistical software or tables).

Let me know if you'd like a more detailed explanation of any of the steps or if you have any other questions.