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Let's analyze and solve each part of this problem using the provided information.

### Given Information:
- **Mean (μ)** of GPAs: 3.15
- **Standard Deviation (σ)**: 0.18
- The GPAs follow a normal distribution.

---

### Solution

#### (a) What percentage of students have GPAs between 2.50 and 3.50?

To find this percentage, calculate the **z-scores** for the GPA values of 2.50 and 3.50:
1. **Z-score for 2.50**:
   \[
   Z = \frac{2.50 - 3.15}{0.18} = \frac{-0.65}{0.18} \approx -3.61
   \]
2. **Z-score for 3.50**:
   \[
   Z = \frac{3.50 - 3.15}{0.18} = \frac{0.35}{0.18} \approx 1.94
   \]

Using a standard normal distribution table or a calculator:
- The area (probability) corresponding to \( Z = -3.61 \) is close to 0%.
- The area for \( Z = 1.94 \) is approximately 97.4%.

So, the percentage of students with GPAs between 2.50 and 3.50 is approximately:
\[
97.4\% - 0\% \approx 97\%
\]

**Answer (a):** 97%

---

#### (b) Halona has a GPA of 3.15. What is Halona's percentile?

Since 3.15 is the mean GPA, Halona's score corresponds to the 50th percentile in a normal distribution.

**Answer (b):** 50th percentile

---

#### (c) What GPA corresponds to Quartile 1?

For a normal distribution, the first quartile (Q1) is the 25th percentile. To find the GPA corresponding to this, calculate the z-score for the 25th percentile:
1. The **z-score** for the 25th percentile is approximately **-0.675**.
2. Using this z-score to find the corresponding GPA:
   \[
   GPA = \mu + (Z \cdot \sigma) = 3.15 + (-0.675 \times 0.18) \approx 3.15 - 0.1215 = 3.03
   \]

**Answer (c):** 3.03

---

#### (d) What GPA is needed to be in the top 5%?

The top 5% corresponds to the 95th percentile. To find this GPA:
1. The **z-score** for the 95th percentile is approximately **1.645**.
2. Calculate the GPA using this z-score:
   \[
   GPA = \mu + (Z \cdot \sigma) = 3.15 + (1.645 \times 0.18) \approx 3.15 + 0.2961 = 3.45
   \]

**Answer (d):** 3.45

---

### Summary of Answers
- (a) **97%**
- (b) **50th percentile**
- (c) **3.03**
- (d) **3.45**

---

Would you like further details on any part of this solution?

Here are five related questions that build on these concepts:
1. How would the answers change if the standard deviation was larger?
2. What percentage of students have GPAs above 3.50?
3. How would you calculate the GPA corresponding to Quartile 3 (75th percentile)?
4. What is the probability of a student having a GPA between 3.15 and 3.50?
5. How would we calculate the GPA needed for the top 1% of students?

**Tip:** For normal distribution problems, always start by finding the z-score to convert values into standard units.

The student GPAs at a certain college are normally distributed with mean of 3.15 and standard deviation 0.18. (a) What percentage of students from this college have GPAs between 2.50 and 3.50? (b) Halona has a GPA of 3.15. What is Halona's percentile? (c) What GPA corresponds to Quartile 1? (d) What GPA is needed to be in the top 5%?

Solve problems related to student GPAs at a college, which follow a normal distribution with a mean of 3.15 and a standard deviation of 0.18. Learn how to calculate the percentage of students within a GPA range, determine percentiles, and identify GPAs for specific quartiles and top percentages. Suitable for college-level statistics.

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