To calculate the regression line $y = mx + b$, we need to compute the slope $m$ and the y-intercept $b$ using the following formulas:

$m = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2}$ $b = \frac{\sum y_i - m \sum x_i}{n}$

Where:

• $n$ is the number of data points.
• $x_i$ and $y_i$ are the values of $x$ and $y$, respectively.

Let me extract the data from the image and perform these calculations. numpy as np

Given data points from the image

x = np.array([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]) y = np.array([19.4, 19.35, 16.9, 16.75, 19.4, 18.35, 21.5, 21.05, 18.5, 19.85, 21.2, 20.45])

Number of data points

n = len(x)

Calculating the slope (m)

m = (n * np.sum(x * y) - np.sum(x) * np.sum(y)) / (n * np.sum(x**2) - (np.sum(x)**2))

Calculating the y-intercept (b)

b = (np.sum(y) - m * np.sum(x)) / n

m, bThe regression line is:

$y = 0.23x + 17.67$

Would you like a detailed breakdown of the calculations or any further explanation?

Here are 5 related questions:

1. How do we interpret the slope in this regression line?
2. What does the y-intercept signify in this context?
3. How can we calculate the correlation coefficient to measure the strength of this regression?
4. Can the regression line predict temperatures for values of $x$ outside the given range?
5. What assumptions are made when calculating a linear regression?

Tip: Always check the residuals (differences between observed and predicted values) to assess the fit of the regression line!