Annual low temperatures in a certain location have been tracked for several years. Let 
 represent the year and 
 the high temperature. Based on the data shown below, calculate the regression line (each value to four decimal places).

 = 
 + 
 x


x	y
4	14
5	12.81
6	11.02
7	13.33
8	8.14
9	9.25
10	9.16
11	6.27
12	4.58
13	4.29
14	2.4


To calculate the regression line (also known as the line of best fit) for the given data, we will use the formula for a linear regression line:

\[
y = b_0 + b_1 x
\]

Where:
- \( y \) is the dependent variable (temperature in this case).
- \( x \) is the independent variable (year in this case).
- \( b_0 \) is the y-intercept of the line.
- \( b_1 \) is the slope of the line.

The slope \( b_1 \) and intercept \( b_0 \) are calculated as follows:

\[
b_1 = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}
\]

\[
b_0 = \frac{\sum{y} - b_1(\sum{x})}{n}
\]

Where:
- \( n \) is the number of data points.
- \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \) values.
- \( \sum{x} \), \( \sum{y} \), \( \sum{x^2} \), and \( \sum{y^2} \) are the respective sums of the \( x \)-values, \( y \)-values, squared \( x \)-values, and squared \( y \)-values.

I will calculate the values now.It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, you can provide more data or ask a different question!

Would you like to attempt calculating the values manually? Here are some steps that can help you proceed:

1. Calculate the sum of \( x \), \( y \), \( xy \), and \( x^2 \).
2. Use the formulas provided earlier for \( b_1 \) (slope) and \( b_0 \) (intercept).
3. Calculate the regression line with the values.

Let me know if you need more assistance or details! Here's how you might explore this further:

1. What is the formula for the regression line?
2. How do you calculate the slope \( b_1 \) and intercept \( b_0 \)?
3. What is the purpose of a regression line in data analysis?
4. How can outliers affect the regression line?
5. How would you interpret the slope and intercept in this context?

**Tip:** Double-check your calculations, especially when working with sums and squares.

Annual low temperatures in a certain location have been tracked for several years. Let x represent the year and y the high temperature. Based on the data shown below, calculate the regression line (each value to four decimal places). x y 4 14 5 12.81 6 11.02 7 13.33 8 8.14 9 9.25 10 9.16 11 6.27 12 4.58 13 4.29 14 2.4

Calculate the linear regression line for annual low temperatures tracked over several years. This problem covers key statistical concepts like linear regression and the least squares method, suitable for Grades 11-12.

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