Math Problem Statement

Find the least-squares regression line treating the commute time, x, as the explanatory variable and the index score, y, as the response variable.

Solution

To solve this problem, we need to compute the least-squares regression line, which has the general form:

y^=b1x+b0\hat{y} = b_1x + b_0

Where:

  • y^\hat{y} is the predicted value of the response variable (Well-Being Index Score),
  • xx is the explanatory variable (Commute Time),
  • b1b_1 is the slope of the regression line,
  • b0b_0 is the y-intercept.

The formulas to calculate the slope (b1b_1) and y-intercept (b0b_0) are as follows:

b1=nxyxynx2(x)2b_1 = \frac{n\sum{xy} - \sum{x}\sum{y}}{n\sum{x^2} - (\sum{x})^2} b0=yb1xnb_0 = \frac{\sum{y} - b_1\sum{x}}{n}

Where:

  • nn is the number of data points,
  • xx is the Commute Time data,
  • yy is the Well-Being Index Score data.

I'll begin by calculating these values based on the data points provided in the image:

  • Commute Time xx: 5, 20, 25, 40, 60, 84, 105
  • Well-Being Index Score yy: 68.9, 67.2, 66.4, 65.1, 63.3, 61.8, 59.3

Let me compute these values.The least-squares regression line, treating the commute time (xx) as the explanatory variable and the well-being index score (yy) as the response variable, is:

y^=0.091x+68.987\hat{y} = -0.091x + 68.987

This equation shows that for every additional minute of commute time, the well-being index score decreases by approximately 0.091 points.

Would you like more details or have any questions?

Here are five related questions to expand on this concept:

  1. How do you interpret the y-intercept in the context of this problem?
  2. How much does the well-being score change if the commute time increases by 10 minutes?
  3. What would the predicted well-being score be for a commute time of 50 minutes?
  4. How would you calculate the coefficient of determination (R2R^2) for this regression model?
  5. What assumptions underlie the use of the least-squares regression method?

Tip: When analyzing regression results, it's important to check whether the data shows a linear relationship before fully trusting the regression line.

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Math Problem Analysis

Mathematical Concepts

Least-Squares Regression
Linear Regression
Algebra

Formulas

ŷ = b1x + b0
b1 = (n * Σxy - Σx * Σy) / (n * Σx^2 - (Σx)^2)
b0 = (Σy - b1 * Σx) / n

Theorems

Least-Squares Method

Suitable Grade Level

Grades 10-12