find the least squares line: (1,16.7) (3,26) (5,33.3) (7,48.3) (9,57) (11,65.8) (13,74.2) (15,83.3)

To find the least squares line (also known as the line of best fit) for the given set of points, we can use the formulas for the slope \( m \) and the y-intercept \( b \) of the line \( y = mx + b \). The formulas are 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,
- \( x \) and \( y \) are the coordinates of the points,
- \( \sum{xy} \) is the sum of the product of each pair of \( x \) and \( y \),
- \( \sum{x^2} \) is the sum of the squares of each \( x \),
- \( \sum{x} \) is the sum of all \( x \) values, and
- \( \sum{y} \) is the sum of all \( y \) values.

Given the points \((1, 16.7)\), \((3, 26)\), \((5, 33.3)\), \((7, 48.3)\), \((9, 57)\), \((11, 65.8)\), \((13, 74.2)\), and \((15, 83.3)\), let's calculate \( m \) and \( b \).

I'll perform the calculations step by step.The least squares line that best fits the given data points is:

\[
y = 4.842x + 11.842
\]

This equation suggests that the slope \( m \) is approximately \( 4.842 \) and the y-intercept \( b \) is approximately \( 11.842 \).

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you interpret the slope and intercept of the least squares line?
2. What are the properties of the least squares line?
3. How would you calculate the residuals for this line?
4. How can you use the least squares line to make predictions?
5. What is the correlation coefficient, and how is it related to the least squares line?
6. How does the sum of squared residuals relate to the least squares method?
7. What is the meaning of a positive vs. a negative slope in a linear regression?
8. How does the choice of data points affect the least squares line?

**Tip:** When interpreting the slope of a least squares line, remember that it represents the change in the dependent variable (y) for a one-unit change in the independent variable (x).

Explore how to find the least squares line for the given set of data points (1, 16.7), (3, 26), (5, 33.3), (7, 48.3), (9, 57), (11, 65.8), (13, 74.2), and (15, 83.3). This page provides a step-by-step solution using linear regression and the least squares method, covering the formulas for slope and y-intercept. Suitable for high school students studying linear regression.

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