Year

Consumer

Credit

Year

Consumer

Credit
 
20042004

2250.52250.5

20092009

2551.52551.5

20052005

2319.82319.8

20102010

2648.12648.1

20062006

2437.02437.0

20112011

2757.02757.0

20072007

2540.92540.9

20122012

2926.32926.3

20082008

2592.12592.1

20132013

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 

20002000.

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, we need to first define the variables:

- Let \( x \) represent the number of years since 2000. Therefore, for each year given, \( x \) is calculated as \( x = \text{Year} - 2000 \).
- Let \( y \) represent the consumer credit for each corresponding year.

### Step 1: Convert the Years to \( x \)-values
Convert the given years into the number of years since 2000:

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

### Step 2: List the Corresponding Consumer Credit Values
The corresponding \( y \)-values (consumer credit) for each \( x \)-value:

\[
\begin{aligned}
x = 4, & \quad y = 2250.5 \\
x = 5, & \quad y = 2319.8 \\
x = 6, & \quad y = 2437.0 \\
x = 7, & \quad y = 2540.9 \\
x = 8, & \quad y = 2592.1 \\
x = 9, & \quad y = 2551.5 \\
x = 10, & \quad y = 2648.1 \\
x = 11, & \quad y = 2757.0 \\
x = 12, & \quad y = 2926.3 \\
x = 13, & \quad y = 3094.2 \\
\end{aligned}
\]

### Step 3: Calculate the Least Squares Line Equation
The least squares line equation is generally given by:

\[
y = mx + b
\]

where \( m \) (the slope) and \( b \) (the y-intercept) are calculated as follows:

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

Where:
- \( n \) is the number of data points
- \( \sum xy \) is the sum of the products of corresponding \( x \) and \( y \) values
- \( \sum x \) 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's calculate these values. I'll do this in steps and then calculate the values of \( m \) and \( b \).

### Step 4: Compute Necessary Sums
I'll compute the necessary sums using the data provided.

\[
\sum x = 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 85
\]

\[
\sum y = 2250.5 + 2319.8 + 2437.0 + 2540.9 + 2592.1 + 2551.5 + 2648.1 + 2757.0 + 2926.3 + 3094.2 = 26117.4
\]

\[
\sum xy = 4(2250.5) + 5(2319.8) + 6(2437.0) + 7(2540.9) + 8(2592.1) + 9(2551.5) + 10(2648.1) + 11(2757.0) + 12(2926.3) + 13(3094.2) 
\]

\[
\sum x^2 = 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 + 13^2 = 819
\]

I'll calculate these values and then compute \( m \) and \( b \).

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

\[
y = 83.15x + 1904.96
\]

This equation represents the relationship between the number of years since 2000 (\( x \)) and the consumer credit (\( y \)).

Would you like more details on how the equation was derived, or do you have any further questions?

Here are five related questions:
1. How do you interpret the slope in the context of this problem?
2. What does the y-intercept represent in this scenario?
3. How would you use this equation to predict consumer credit for the year 2015?
4. What are the limitations of using a least squares line for this data?
5. How could you assess the accuracy of this linear model?

**Tip:** When interpreting the slope in a linear equation, consider what the change in \( y \) represents for every one-unit increase in \( x \).

Explore the calculation of the least squares line equation for consumer credit data over years since 2000. Understand how linear regression is applied to predict consumer credit trends. Detailed solution and analysis provided.

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