AP Precalculus Practice Test: Unit 2 Question #15 Residual Plots and Appropriate Linear Models

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

### AP Precalculus Practice Test: Unit 2, Question #15
**Topic:** Residual Plots and Appropriate Linear Models

This question involves using residual plots to determine if a linear model is appropriate for a set of data. Residual plots show the differences (residuals) between observed and predicted values. By analyzing the plot, you can assess the fit of a linear model.

---

### **Steps to Analyze Residual Plots:**

#### 1. **Residuals:**
- A **residual** is the difference between the observed value and the predicted value:
\[
\text{Residual} = y - \hat{y}
\]
- Residuals measure how well the model fits the data.

#### 2. **Creating a Residual Plot:**
- Perform linear regression, calculate predicted values, then subtract them from observed values to get residuals. Plot the residuals against the corresponding \(x\)-values.

#### 3. **Interpreting the Residual Plot:**
- **Good fit**: Random scatter around the horizontal axis, indicating the linear model is appropriate.
- **Poor fit**: A pattern in the residuals suggests the need for a different model (e.g., quadratic or exponential).

#### 4. **TI-84+ Steps:**
- Perform linear regression, calculate residuals, and graph them using the **`STAT PLOT`** feature on the calculator.

#### 5. **Example Problem:**
Given data points, perform linear regression, calculate residuals, and analyze the residual plot to see if the linear model is a good fit.

---

### **Practice Problem:**
Given the data set:

| \(x\) | \(y\) |
|------|------|
| 1 | 5 |
| 2 | 10 |
| 3 | 15 |
| 4 | 20 |
| 5 | 30 |

Perform linear regression, calculate residuals, and analyze the residual plot to assess if a linear model is appropriate.

---

**Key Takeaways:**
- **Random scatter** in the residual plot suggests a good linear model fit.
- **Patterned residuals** suggest a different model might be needed.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

/ nickperich

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

#math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study