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To solve this problem using **linear classification**, we will follow these steps:

### 1. **Understanding the Data**
We are given a set of exam scores from two exams for five students, along with their pass/fail outcomes. The goal is to predict whether the new student, with exam scores \( 70 \) (Exam 1) and \( 65 \) (Exam 2), will pass or fail.

Here’s the table:

| Student | Exam 1 Score | Exam 2 Score | Pass/Fail |
|---------|--------------|--------------|-----------|
| A       | 80           | 85           | Pass      |
| B       | 90           | 75           | Pass      |
| C       | 50           | 60           | Fail      |
| D       | 60           | 70           | Fail      |
| E       | 75           | 80           | Pass      |
| New     | 70           | 65           | ?         |

### 2. **Formulating the Problem**
This is a binary classification problem where the decision boundary is linear. We'll use the given exam scores as two features, \( x_1 \) (Exam 1 score) and \( x_2 \) (Exam 2 score). The labels are either "Pass" or "Fail".

We aim to find a linear decision boundary (a straight line) in the feature space that separates the students who passed from those who failed.

### 3. **Setting Up a Linear Classifier**
The equation for a line (decision boundary) in two dimensions is:

\[
w_1 x_1 + w_2 x_2 + b = 0
\]

Where:
- \( w_1 \) and \( w_2 \) are weights,
- \( x_1 \) and \( x_2 \) are the features (exam scores),
- \( b \) is the bias term (intercept).

Our goal is to find the values of \( w_1 \), \( w_2 \), and \( b \) that best separate the passing and failing students.

### 4. **Plotting the Data**
We can start by plotting the data points and visually inspecting if a linear boundary can be drawn. Here are the student scores:

- Pass: \( (80, 85) \), \( (90, 75) \), \( (75, 80) \)
- Fail: \( (50, 60) \), \( (60, 70) \)

And for the new student, the point is \( (70, 65) \).

### 5. **Linear Separation**
We can now estimate a decision boundary by observing the data distribution. If we assume the decision boundary is a line, we will attempt to separate the two groups (Pass and Fail). A reasonable choice for the line based on the data could be:

\[
x_2 = x_1 - 10
\]

This means:
- If \( x_2 \geq x_1 - 10 \), the student will pass.
- If \( x_2 < x_1 - 10 \), the student will fail.

### 6. **Predicting for the New Student**
The new student has the following scores:
- \( x_1 = 70 \)
- \( x_2 = 65 \)

We apply the decision rule:

\[
x_2 = 65 \quad \text{and} \quad x_1 - 10 = 70 - 10 = 60
\]

Since \( 65 > 60 \), according to the rule \( x_2 \geq x_1 - 10 \), the new student is predicted to **pass**.

---

### Conclusion
The new student is predicted to **Pass** based on the linear classification.

Would you like more details on how to improve this classifier, or any questions on specific steps? 

Here are some related questions for deeper exploration:
1. How would the decision boundary change if we used a different method to classify the students?
2. Can we adjust the weights to get a more precise classifier?
3. What if we had more features, such as attendance or homework scores?
4. How would non-linear classification work in this scenario?
5. How do we measure the accuracy of this linear classification?

**Tip:** In real-world applications, it's important to test your classifier on unseen data to ensure it generalizes well.

Given a dataset of students' exam scores and their corresponding pass/fail status, use linear classification to predict whether a new student with certain exam scores will pass or fail.

Predict whether a new student will pass or fail based on their exam scores using a linear classifier. This example uses data from five students and applies a linear decision boundary. Suitable for undergraduate-level courses in machine learning and binary classification.

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