To find the linear regression line $y = mx + b$ using the given data, we need to compute the slope ($m$) and y-intercept ($b$) using the 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,
• $\sum x$ is the sum of $x$-values,
• $\sum y$ is the sum of $y$-values,
• $\sum xy$ is the sum of the product of $x$ and $y$-values,
• $\sum x^2$ is the sum of the squares of $x$-values.

Let’s calculate these values step by step. Data from the table x_values = [0, 1, 2, 3, 4, 5] y_values = [2.12, 2.19, 1.92, 2.79, 3.81, 4.72]

Number of data points

n = len(x_values)

Calculations for summations

sum_x = sum(x_values) sum_y = sum(y_values) sum_x2 = sum(x ** 2 for x in x_values) sum_xy = sum(x * y for x, y in zip(x_values, y_values))

Slope (m) and intercept (b)

m = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x ** 2) b = (sum_y - m * sum_x) / n

Round the results to 2 decimal places

m_rounded = round(m, 2) b_rounded = round(b, 2)

m_rounded, b_roundedThe equation of the linear regression line is:

$y = 0.54x + 1.59$

Would you like to see detailed calculations or have any questions? Here are 5 related questions to explore further:

1. How is the coefficient of determination ($R^2$) calculated for this regression line?
2. What are the practical interpretations of $m$ (slope) and $b$ (intercept) in this context?
3. Can you predict the value of $y$ when $x = 6$ using this regression line?
4. What would happen if one of the data points was an outlier? How would it affect the line?
5. How can residuals be used to assess the quality of the regression line?

Tip: Always plot the data and the regression line to visualize the fit.