Calculating Sum of Squared Errors (SSE) for Car Stopping Distance Predictions

Math Problem Statement

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.

speed (x)

distance (y)

predicted (ŷ = 7x - 100)

residual (y - ŷ)

squared residual

(ŷ - y)^2

110

484

352

320

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 = 8x - 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 = 8x - 120)

residual (y - ŷ)

squared residual

(ŷ - y)^2

120

1024

352

360

Add the last column to get the SSE for Bellatrix:

Solution

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?

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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Regression
Residuals
Sum of Squared Errors (SSE)

Formulas

ŷ = 7x - 100 (Albus' equation)
ŷ = 8x - 120 (Bellatrix' equation)
Residual = y - ŷ
Squared Residual = (y - ŷ)^2
Sum of Squared Errors (SSE) = Σ(y - ŷ)^2

Theorems

Least Squares Regression

Suitable Grade Level

Grades 10-12

Related Recommendation

Calculate Sum of Squared Errors (SSE) - Statistical Accuracy Measure

Sum of Squared Errors (SSE) Calculation for Regression Model

Compute Sum of Squares Error (SSE) for Data Points and Line y = x + 1

Sum of Squares Error (SSE) Calculation for Linear Model y = x + 1

Calculate the Sum of Squared Errors (SSE) in Linear Regression Model for College and High School GPAs