20042004

2219.52219.5

20092009

2553.52553.5

20052005

2303.82303.8

20102010

2648.12648.1

20062006

2465.02465.0

20112011

2757.02757.0

20072007

2517.92517.9

20122012

2926.32926.3

20082008

2597.12597.1

20132013

3099.23099.2

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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 for the least squares line, we need to determine the linear regression equation, which is in the form:

\[
Y = mx + b
\]

where:
- \( Y \) is the dependent variable (in this case, the given values for each year).
- \( x \) is the independent variable (the number of years since 2000).
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 1: Determine the variables
Let:
- \( x = \text{Year} - 2000 \) (so \( x \) represents the number of years since 2000)
- \( Y = \text{Value} \) (the corresponding values provided for each year)

Given dataTo find the equation for the least squares line, we need to determine the linear regression equation, which is in the form:

\[
Y = mx + b
\]

where:
- \( Y \) is the dependent variable (in this case, the given values for each year).
- \( x \) is the independent variable (the number of years since 2000).
- \( m \) is the slope of the line.
- \( b \) is the y-intercept.

### Step 1: Determine the variables
Let:
- \( x = \text{Year} - 2000 \) (so \( x \) represents the number of years since 2000)
- \( Y = \text{Value} \) (the corresponding values provided for each year)

Given 

\[
\begin{align*}
\text{Year (x)} & \quad \text{Y} \\
0 \, (2000) & \quad 2219.5 \\
1 \, (2001) & \quad 2303.8 \\
2 \, (2002) & \quad 2465.0 \\
3 \, (2003) & \quad 2517.9 \\
4 \, (2004) & \quad 2597.1 \\
5 \, (2005) & \quad 2648.1 \\
6 \, (2006) & \quad 2757.0 \\
7 \, (2007) & \quad 2926.3 \\
8 \, (2008) & \quad 3099.2 \\
\end{align*}
\]

### Step 2: Calculate the slope (m) and y-intercept (b)

We use the formulas for \( m \) and \( b \):

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

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

where:
- \( n \) = number of data points
- \( \sum x \) = sum of all \( x \) values
- \( \sum y \) = sum of all \( Y \) values
- \( \sum xy \) = sum of the product of each \( x \) and \( Y \)
- \( \sum x^2 \) = sum of each \( x \) squared

Let's compute these values to get the least squares line equation.The equation for the least squares line, rounding the slope and y-intercept to two decimal places, is:

\[
Y = 101.68x + 2208.18
\]

where:
- \( x \) is the number of years since 2000.
- \( Y \) is the predicted value for the corresponding year.

### Do you want more details or have any questions?

### Related Questions:
1. How is the least squares method used to minimize the error in regression?
2. What are other ways to measure the goodness of fit for a regression line?
3. Can you provide the correlation coefficient for this data set?
4. What would the regression line predict for the year 2020?
5. How does the data's spread affect the slope of the regression line?

**Tip:** Remember to always check the residuals to verify the accuracy of your regression model!

Discover how to find the equation for the least squares line using linear regression. Understand the application of the least squares method to calculate the slope and y-intercept from provided data points since the year 2000. Includes step-by-step calculations and formulas.

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