Math Problem Statement

Based on the data shown below, calculate the regression line (each value to two decimal places): x = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] y = [26.52, 27.3, 23.28, 23.66, 21.24, 18.42, 18.5, 16.98, 14.46, 14.64, 12.72, 9.5]

Solution

To solve for the regression line based on the given data, I will calculate the equation of the line y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Let me calculate these values step by step:

  1. Compute the slope mm using the formula: m=n(xy)xyn(x2)(x)2m = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}

  2. Compute the y-intercept bb using: b=ymxnb = \frac{\sum y - m \sum x}{n}

Where nn is the number of data points. Let me calculate this. pandas as pd

Data from the image

data = { 'x': [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], 'y': [26.52, 27.3, 23.28, 23.66, 21.24, 18.42, 18.5, 16.98, 14.46, 14.64, 12.72, 9.5] }

Convert data to a DataFrame

df = pd.DataFrame(data)

Calculate sums needed for the regression line calculation

n = len(df) sum_x = df['x'].sum() sum_y = df['y'].sum() sum_xy = (df['x'] * df['y']).sum() sum_x2 = (df['x']**2).sum()

Calculate slope (m) and intercept (b)

m = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) b = (sum_y - m * sum_x) / n

m, bThe regression line for the given data is:

y=1.53x+30.41y = -1.53x + 30.41

Would you like to see the detailed steps for calculating this, or have additional questions? Here are 5 related questions you might explore:

  1. How is the slope calculated step by step for regression?
  2. How can we interpret the slope and intercept in this context?
  3. How would the regression line change if more data points were added?
  4. Can you provide a plot of the data and the regression line?
  5. How can we use this regression line to predict values for new xx?

Tip: Always check the scatterplot of the data to visually verify the fit of your regression line!

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistics
Algebra

Formulas

Slope formula: m = (nΣ(xy) - ΣxΣy) / (nΣ(x²) - (Σx)²)
Intercept formula: b = (Σy - mΣx) / n

Theorems

Least Squares Method

Suitable Grade Level

Grades 11-12, College