Year

Consumer

Credit

Year

Consumer

Credit
 
20032003

2219.52219.5

20082008

2557.52557.5

20042004

2303.82303.8

20092009

2648.12648.1

20052005

2437.02437.0

20102010

2756.02756.0

20062006

2517.92517.9

20112011

2926.32926.3

20072007

2595.12595.1

20122012

3094.23094.2

Question content area bottom

Part 1

**a**. 

Find an equation for the least squares​ line, letting x equal the number of years since 

20012001.

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 of the least squares line (also known as the line of best fit), we need to follow these steps:

### Step 1: Define the Variables
- Let \( x \) be the number of years since 2001. So, for the year 2003, \( x = 2003 - 2001 = 2 \), and so on.
- Let \( y \) be the consumer credit for each year.

### Step 2: Organize the Data
We'll list the values of \( x \) and \( y \) accordingly:

\[
\begin{array}{|c|c|c|}
\hline
\text{Year} & x & y (\text{Consumer Credit}) \\
\hline
2003 & 2 & 2219.5 \\
2004 & 3 & 2303.8 \\
2005 & 4 & 2437.0 \\
2006 & 5 & 2517.9 \\
2007 & 6 & 2595.1 \\
2008 & 7 & 2557.5 \\
2009 & 8 & 2648.1 \\
2010 & 9 & 2756.0 \\
2011 & 10 & 2926.3 \\
2012 & 11 & 3094.2 \\
\hline
\end{array}
\]

### Step 3: Calculate the Slope (\( m \)) and Y-intercept (\( b \))
The formula for the slope \( m \) of the least squares line is:

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

The formula for the y-intercept \( b \) is:

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

Where \( n \) is the number of data points (in this case, 10).

### Step 4: Perform the Calculations

1. **Sum of \( x \) values**: \( \sum x = 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 65 \)
2. **Sum of \( y \) values**: \( \sum y = 2219.5 + 2303.8 + 2437.0 + 2517.9 + 2595.1 + 2557.5 + 2648.1 + 2756.0 + 2926.3 + 3094.2 = 26055.4 \)
3. **Sum of \( x^2 \)**: \( \sum x^2 = 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 = 505 \)
4. **Sum of \( xy \)**: \( \sum xy = (2 \times 2219.5) + (3 \times 2303.8) + (4 \times 2437.0) + \dots + (11 \times 3094.2) = 176343.2 \)

Now, substitute these into the formulas:

\[
m = \frac{10(176343.2) - (65)(26055.4)}{10(505) - (65)^2}
\]
\[
m = \frac{1763432 - 1693591}{5050 - 4225}
\]
\[
m = \frac{69841}{825} \approx 84.66
\]

Now, calculate \( b \):

\[
b = \frac{(26055.4)(505) - (65)(176343.2)}{5050 - 4225}
\]
\[
b = \frac{13177877 - 11462208}{825}
\]
\[
b = \frac{1715669}{825} \approx 2079.6
\]

### Step 5: Write the Equation
The equation for the least squares line is:

\[
y = 84.66x + 2079.6
\]

This equation can be rounded to:

\[
y = 84.66x + 2079.60
\]

### Final Answer
The equation of the least squares line is:

\[
y = 84.66x + 2079.60
\]

Would you like to see the details of these calculations, or have any questions?

**Related Questions:**
1. How do you interpret the slope of the least squares line in the context of this data?
2. What does the y-intercept represent in this scenario?
3. How would you predict the consumer credit for the year 2013 using this model?
4. What are the potential limitations of using a linear model for this data?
5. How would the equation change if you considered a different base year (e.g., 2000 instead of 2001)?

**Tip:** Always check your work by substituting values back into the equation to ensure the line fits the data points closely.

Learn how to find the equation of the least squares line for consumer credit data from 2003 to 2012 using linear regression. Understand the slope and y-intercept calculations with detailed step-by-step solutions.

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