The total amount of consumer credit has been increasing steadily in recent years. The following table gives the total outstanding consumer credit​ (in billions of​ dollars). Answer parts​ (a) through​ (e) below.
Year
Consumer
Credit
Year
Consumer
Credit

2004
2210.5
2009
2557.5
2005
2303.8
2010
2643.1
2006
2437.0
2011
2759.0
2007
2540.9
2012
2924.3
2008
2597.1
2013
3099.2
Question content area bottom
Part 1
a. Find an equation for the least squares​ line, letting x equal the number of years since 2000.
The equation is Yequals

148.73xplus

1343.04.
​(Use integers or decimals for any numbers in the equation. Round to two decimal places as​ needed.)

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.

Learn how to calculate the least squares line for consumer credit data using the provided yearly values. Understand the slope and y-intercept of the line and its application in predicting consumer credit trends. Suitable for advanced high school students or anyone interested in statistical modeling.

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