To solve this problem and determine the equation of the regression line, I will calculate the slope ($m$) and intercept ($b$) of the line based on the provided $x$ and $y$ data.

The formula for the slope ($m$) is: $m = \frac{n \sum(xy) - \sum x \sum y}{n \sum x^2 - (\sum x)^2}$

The formula for the intercept ($b$) is: $b = \frac{\sum y - m \sum x}{n}$

Where $n$ is the number of data points.

Let me compute this for you. numpy as np import pandas as pd

Data extracted from the image

x = np.array([5, 3, 6, 4, 11, 10, 7, 9, 12, 8, 13]) y = np.array([9.58, 4.14, 11.76, 7.04, 17.38, 16.96, 13.60, 16.20, 17.44, 15.08, 17.14])

Number of data points

n = len(x)

Calculating the sums needed for slope and intercept

sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x**2)

Calculating 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

m, bThe equation of the regression line is:

$\hat{y} = 1.30x + 2.90$

Here, the slope ($m$) is 1.30 and the intercept ($b$) is 2.90 (rounded to two decimal places).

Additional Insights:

This equation allows you to predict $y$ values based on $x$ values using the linear relationship derived from the data.

Would you like details on the scatterplot characteristics or more calculations? Here are some follow-up questions you might explore:

1. How do we interpret the slope and intercept in context?
2. Can we visualize this data with a scatterplot and the regression line?
3. What is the correlation coefficient between $x$ and $y$?
4. Are there any outliers in the data that could impact the regression line?
5. How do we assess the goodness of fit for this regression model?

Tip: Always examine scatterplots to ensure the linear model fits the data appropriately before relying on predictions.