Math Problem Statement
Solution
To solve this problem, we need to compute the least-squares regression line, which has the general form:
Where:
- is the predicted value of the response variable (Well-Being Index Score),
- is the explanatory variable (Commute Time),
- is the slope of the regression line,
- is the y-intercept.
The formulas to calculate the slope () and y-intercept () are as follows:
Where:
- is the number of data points,
- is the Commute Time data,
- 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 : 5, 20, 25, 40, 60, 84, 105
- Well-Being Index Score : 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 () as the explanatory variable and the well-being index score () as the response variable, is:
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:
- How do you interpret the y-intercept in the context of this problem?
- How much does the well-being score change if the commute time increases by 10 minutes?
- What would the predicted well-being score be for a commute time of 50 minutes?
- How would you calculate the coefficient of determination () for this regression model?
- 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
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