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Part 1
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
2250.5
2009
2557.5
2005
2303.8
2010
2645.1
2006
2437.0
2011
2757.0
2007
2540.9
2012
2926.3
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

enter your response herexplus

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

To find the equation for the least squares line, we'll need to perform a linear regression analysis using the given data points. The general form of the equation for a straight line is:

\[
Y = mx + b
\]

Where:
- \( Y \) is the total outstanding consumer credit.
- \( x \) is the number of years since 2000.
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 1: Convert the Years to \( x \)
The table's years are converted to the number of years since 2000:

\[
\begin{array}{|c|c|}
\hline
\text{Year} & x = \text{Year} - 2000 \\
\hline
2004 & 4 \\
2005 & 5 \\
2006 & 6 \\
2007 & 7 \\
2008 & 8 \\
2009 & 9 \\
2010 & 10 \\
2011 & 11 \\
2012 & 12 \\
2013 & 13 \\
\hline
\end{array}
\]

### Step 2: List the Corresponding Consumer Credit Data
Use the values given in the table for the consumer credit (Y):

\[
\begin{array}{|c|c|}
\hline
\text{Year} (x) & \text{Consumer Credit (Y in billions)} \\
\hline
4 & 2250.5 \\
5 & 2303.8 \\
6 & 2437.0 \\
7 & 2540.9 \\
8 & 2597.1 \\
9 & 2557.5 \\
10 & 2645.1 \\
11 & 2757.0 \\
12 & 2926.3 \\
13 & 3099.2 \\
\hline
\end{array}
\]

### Step 3: Calculate the Slope \( m \) and Intercept \( b \)
The formulas for the slope \( m \) and the y-intercept \( b \) are:

\[
m = \frac{N \sum{xy} - \sum{x} \sum{y}}{N \sum{x^2} - (\sum{x})^2}
\]

\[
b = \frac{\sum{y} - m \sum{x}}{N}
\]

Where:
- \( N \) is the number of data points (10 in this case).
- \( \sum{xy} \) is the sum of the product of each \( x \) and \( y \).
- \( \sum{x} \) is the sum of the \( x \) values.
- \( \sum{y} \) is the sum of the \( y \) values.
- \( \sum{x^2} \) is the sum of the squares of the \( x \) values.

Let me calculate these values to find the equation.The equation for the least squares line is:

\[
Y = 84.05x + 1896.99
\]

Here:
- The slope \( m \) is approximately \( 84.05 \).
- The y-intercept \( b \) is approximately \( 1896.99 \).

Would you like to explore further details or ask any questions? Here are some related questions:

1. How can you interpret the slope of this line in the context of consumer credit?
2. What does the y-intercept represent in this context?
3. How would you predict the consumer credit for a year not in the dataset, like 2015?
4. How would the equation change if we used a different base year, like 2005, instead of 2000?
5. What would the sum of squared errors (SSE) be for this least squares line?

**Tip:** When interpreting the slope in the context of a real-world problem, consider what it represents in terms of rate of change. Here, it reflects the annual increase in consumer credit.

Learn how to calculate the least squares line for consumer credit trends based on annual data from 2004 to 2013. Understand the slope and y-intercept of the line and interpret their meaning in the context of consumer credit growth.

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