Let's solve the problem step by step to understand how the equation of the least squares line is derived.

Step 1: Assigning Values for $x$

We are asked to let $x$ be the number of years since 2000. So, for each given year, $x$ can be calculated as:

• 2004: $x = 4$
• 2005: $x = 5$
• 2006: $x = 6$
• 2007: $x = 7$
• 2008: $x = 8$
• 2009: $x = 9$
• 2010: $x = 10$
• 2011: $x = 11$
• 2012: $x = 12$
• 2013: $x = 13$

Step 2: Setting Up the Regression Model

The equation of the least squares line is given by:

$Y = mx + b$

Where:

• $Y$ is the consumer credit (in billions of dollars),
• $x$ is the number of years since 2000,
• $m$ is the slope of the line,
• $b$ is the y-intercept.

Step 3: Finding the Slope $m$ and Y-intercept $b$

The formulas to compute $m$ and $b$ are:

$m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$

$b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2}$

However, as per the problem statement, the equation is already provided:

$Y = 148.73x + 1343.04$

Step 4: Interpretation

• Slope $m = 148.73$: This value means that for each year after 2000, the consumer credit increases by approximately 148.73 billion dollars.
• Y-intercept $b = 1343.04$: This is the estimated consumer credit in the year 2000, which is approximately 1343.04 billion dollars.

Thus, the equation of the least squares line is:

$Y = 148.73x + 1343.04$

This equation models the relationship between the year (represented by $x$) and the total outstanding consumer credit (in billions of dollars).

Would you like more details on how this equation is applied, or do you have any other questions?

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Related Questions:

1. How is the slope of the least squares line interpreted in the context of consumer credit data?
2. What does the y-intercept of the least squares line represent in this scenario?
3. How can this model be used to predict consumer credit in the future, say for the year 2015?
4. What factors could cause deviations from the predictions made by this model?
5. How does the goodness-of-fit (R-squared) influence the reliability of the least squares line in predictions?

Tip:

When interpreting a regression line, always consider the context of the data and whether the model assumptions are reasonable over the range of the data used.