Math Problem Statement

Use the advertised prices for used cars of a particular model to create a linear regression model relating the car's age to its price. Find the equation of the line of regression.

Solution

To determine the equation of the line of regression (Price = b0+b1Ageb_0 + b_1 \cdot \text{Age}), we need to calculate:

  1. The slope (b1b_1): b1=Covariance of Age and PriceVariance of Ageb_1 = \frac{\text{Covariance of Age and Price}}{\text{Variance of Age}}

  2. The y-intercept (b0b_0): b0=yˉb1xˉb_0 = \bar{y} - b_1 \cdot \bar{x} where xˉ\bar{x} is the mean of Age, and yˉ\bar{y} is the mean of Price.

I’ll compute this step by step using the provided data:

Data Points:

  • Age (xx): 1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10
  • Price (yy): 17619, 14959, 15998, 14008, 14988, 14628, 14018, 12019, 10028, 11528, 10828, 9899

Let me calculate this. numpy as np import pandas as pd

Data

age = np.array([1, 2, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10]) price = np.array([17619, 14959, 15998, 14008, 14988, 14628, 14018, 12019, 10028, 11528, 10828, 9899])

Mean of Age and Price

mean_age = np.mean(age) mean_price = np.mean(price)

Slope (b1)

numerator = np.sum((age - mean_age) * (price - mean_price)) denominator = np.sum((age - mean_age) ** 2) slope = numerator / denominator

Intercept (b0)

intercept = mean_price - slope * mean_age

Linear equation components rounded to nearest integer

slope_rounded = round(slope) intercept_rounded = round(intercept)

slope_rounded, intercept_roundedThe equation of the regression line is:

Price=17946818Age\text{Price} = 17946 - 818 \cdot \text{Age}

Would you like a detailed explanation of how this was calculated or assistance with additional parts of the problem?

Here are 5 related questions to consider:

  1. How do we interpret the slope (818-818) in this context?
  2. How accurate is the regression line in predicting prices? (Would you like to calculate R2R^2?)
  3. Can we use this model for predicting prices of cars older than 10 years?
  4. What does the intercept (1794617946) represent?
  5. How can we visualize this linear regression on a graph?

Tip: Always check the scatterplot of your data to ensure the linear regression is a good fit!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Algebra

Formulas

b1 = Σ((x - x̄) * (y - ȳ)) / Σ((x - x̄)^2)
b0 = ȳ - b1 * x̄
Price = b0 + b1 * Age

Theorems

Least Squares Method

Suitable Grade Level

Grades 11-12 or College Level