The distance a car travels after braking is measured.

A car initially traveling at 15 mph travels 22 feet before stopping.

A car initially traveling at 30 mph travels 88 feet before stopping.

A car initially traveling at 60 mph travels 352 feet before stopping.




(a) Albus thinks the best-fitting linear equation for predicting stopping distance based on speed of car is *ŷ* = 7*x* - 100. Complete the following table to compute the sum of squared residuals (sum of squared errors, SSE) for Albus' equation.




speed (*x*)

distance (*y*)

predicted (*ŷ* = 7*x* - 100)

residual (*y* - *ŷ*)

squared residual 

(*ŷ* - y)^2

15

22










30

88

110

- 22

484

60

352

320

32

1024

Add the last column to get the SSE for Albus: 




(b) Bellatrix thinks the best-fitting linear equation for predicting stopping distance based on speed of car is *ŷ* = 8*x* - 120. Complete the following table to compute the sum of squared residuals (sum of squared errors, SSE) for Bellatrix' equation.




speed (*x*)

distance (*y*)

predicted (*ŷ* = 8*x* - 120)

residual (*y* - *ŷ*)

squared residual 

(*ŷ* - y)^2

15

22










30

88

120

- 32

1024

60

352

360

- 8

64

Add the last column to get the SSE for Bellatrix: 







(c) Based on the SSE, whose equation was m

Let's break down the solution step by step to compute the sum of squared residuals (SSE) for both Albus' and Bellatrix's equations.

### Part (a): Albus' Equation (\(\hat{y} = 7x - 100\))

#### Completing the table for Albus:

- **Given equation:** \(\hat{y} = 7x - 100\)
  
| Speed (*x*) | Distance (*y*) | Predicted (\(\hat{y}\)) | Residual (\(y - \hat{y}\)) | Squared Residual (\((y - \hat{y})^2\)) |
|-------------|----------------|-------------------------|----------------------------|----------------------------------------|
| 15          | 22             | \(7(15) - 100 = 105 - 100 = 5\)    | \(22 - 5 = 17\)            | \(17^2 = 289\)                         |
| 30          | 88             | \(7(30) - 100 = 210 - 100 = 110\)  | \(88 - 110 = -22\)         | \((-22)^2 = 484\)                      |
| 60          | 352            | \(7(60) - 100 = 420 - 100 = 320\)  | \(352 - 320 = 32\)         | \(32^2 = 1024\)                        |

- **Sum of Squared Residuals (SSE) for Albus:**
  \[
  289 + 484 + 1024 = 1797
  \]

### Part (b): Bellatrix's Equation (\(\hat{y} = 8x - 120\))

#### Completing the table for Bellatrix:

- **Given equation:** \(\hat{y} = 8x - 120\)
  
| Speed (*x*) | Distance (*y*) | Predicted (\(\hat{y}\)) | Residual (\(y - \hat{y}\)) | Squared Residual (\((y - \hat{y})^2\)) |
|-------------|----------------|-------------------------|----------------------------|----------------------------------------|
| 15          | 22             | \(8(15) - 120 = 120 - 120 = 0\)     | \(22 - 0 = 22\)            | \(22^2 = 484\)                         |
| 30          | 88             | \(8(30) - 120 = 240 - 120 = 120\)   | \(88 - 120 = -32\)         | \((-32)^2 = 1024\)                     |
| 60          | 352            | \(8(60) - 120 = 480 - 120 = 360\)   | \(352 - 360 = -8\)         | \((-8)^2 = 64\)                        |

- **Sum of Squared Residuals (SSE) for Bellatrix:**
  \[
  484 + 1024 + 64 = 1572
  \]

### Part (c): Comparing the SSE
- **Albus' SSE:** 1797
- **Bellatrix's SSE:** 1572

Since Bellatrix's SSE (1572) is **lower** than Albus' SSE (1797), Bellatrix's equation provides a better fit for the data based on the sum of squared errors (SSE).

---

### Do you want further details or have any questions?

### Related Questions:
1. What is the formula for calculating the residual?
2. How does the sum of squared residuals (SSE) help determine the accuracy of a model?
3. Can you explain how least squares regression works in more detail?
4. What is the difference between linear and non-linear regression in modeling data?
5. How do outliers affect the sum of squared residuals?

### Tip:
The sum of squared residuals (SSE) is a useful metric to measure how well a model fits data—lower values indicate a better fit.

The distance a car travels after braking is measured. A car initially traveling at 15 mph travels 22 feet before stopping. A car initially traveling at 30 mph travels 88 feet before stopping. A car initially traveling at 60 mph travels 352 feet before stopping.

(a) Albus thinks the best-fitting linear equation for predicting stopping distance based on speed of car is ŷ = 7x - 100. Complete the following table to compute the sum of squared residuals (sum of squared errors, SSE) for Albus' equation.

(b) Bellatrix thinks the best-fitting linear equation for predicting stopping distance based on speed of car is ŷ = 8x - 120. Complete the following table to compute the sum of squared residuals (sum of squared errors, SSE) for Bellatrix' equation.

(c) Based on the SSE, whose equation was more accurate?

Learn how to calculate the Sum of Squared Errors (SSE) for predicting car stopping distances. This problem involves linear regression and residual calculations using equations ŷ = 7x - 100 and ŷ = 8x - 120. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation