Calculate the least squares line.           x: 1,2,3,4,5,6,7,8,9,10   y:0.3,0.9,1.3,2.3,2.8,3.3,3.3,4.6,4.8,5.3

To calculate the least squares line, we need to determine the linear equation of the form:

\[
y = mx + b
\]

where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

Given the data points:

- \( x: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \)
- \( y: 0.3, 0.9, 1.3, 2.3, 2.8, 3.3, 3.3, 4.6, 4.8, 5.3 \)

We'll calculate the slope \( m \) and the y-intercept \( b \) using the following formulas:

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

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

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

Let's calculate these step by step.The equation of the least squares line is:

\[
y = 0.559x - 0.187
\]

This line best fits the given data points using the least squares method.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you calculate the correlation coefficient for these data points?
2. What is the residual sum of squares (RSS) for this line?
3. How would the least squares line change if we added more data points?
4. Can you explain the significance of the slope and intercept in this context?
5. How does the least squares method minimize error?

**Tip:** The least squares method is widely used in regression analysis because it minimizes the sum of the squares of the residuals, providing the best linear fit.

Learn how to calculate the least squares line using a detailed step-by-step solution. This problem involves statistics and regression analysis, focusing on finding the best-fit line for given data points. Suitable for advanced high school students and beyond.

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