Math Problem Statement
Find an equation for the least squares line, letting x equal the number of years since 2000, given the table of consumer credit values from 2004 to 2012.
Solution
The problem asks for finding an equation of the least squares line based on the provided data, where represents the number of years since 2000, and represents the consumer credit in billions of dollars.
The table gives the following dataThe problem asks for finding an equation of the least squares line based on the provided data, where represents the number of years since 2000, and represents the consumer credit in billions of dollars.
The table gives the following
\hline \text{Year} & \text{Consumer Credit (in billions)} \\ \hline 2004 & 2210.5 \\ 2005 & 2319.8 \\ 2006 & 2466.0 \\ 2007 & 2517.9 \\ 2009 & 2557.5 \\ 2010 & 2648.1 \\ 2011 & 2757.0 \\ 2012 & 2924.3 \\ \hline \end{array}$$ The first step is to convert the "Year" into $$x$$ values where $$x = \text{Year} - 2000$$. This gives us the following points: $$\begin{array}{|c|c|} \hline x & Y \\ \hline 4 & 2210.5 \\ 5 & 2319.8 \\ 6 & 2466.0 \\ 7 & 2517.9 \\ 9 & 2557.5 \\ 10 & 2648.1 \\ 11 & 2757.0 \\ 12 & 2924.3 \\ \hline \end{array}$$ We can now use these points to find the least squares line equation $$Y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept. The formula for the slope $$m$$ is: $$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$ And the y-intercept $$b$$ is: $$b = \frac{\sum y - m(\sum x)}{n}$$ Let me calculate that for you.The equation for the least squares line is: $$Y = 76.18x + 1940.72$$ Here, $$x$$ represents the number of years since 2000, and $$Y$$ is the consumer credit in billions of dollars. Do you need further explanation on how these calculations were done, or have any other questions? Here are five related questions to explore further: 1. What does the slope of $$76.18$$ indicate about the rate of change in consumer credit over time? 2. How would you predict the consumer credit for the year 2025 using this equation? 3. Can you determine the residuals for each data point using this least squares line? 4. What is the interpretation of the y-intercept $$1940.72$$? 5. How would the equation change if more recent data were added to the dataset? **Tip:** The slope of a least squares regression line gives the average change in the dependent variable (consumer credit) for each one-unit increase in the independent variable (years since 2000).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Statistics
Linear Regression
Formulas
Least squares line formula: Y = mx + b
Slope formula: m = (n(∑xy) - (∑x)(∑y)) / (n(∑x^2) - (∑x)^2)
Y-intercept formula: b = (∑y - m(∑x)) / n
Theorems
Least Squares Regression Line
Suitable Grade Level
Grades 9-12