https://file.aiquickdraw.com/mathbot/file/1732911406471soce0umg.jpg

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.

Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

Learn how to calculate the regression line equation using given x and y data points. This tutorial covers key concepts in statistics, including slope and intercept calculation, scatterplot interpretation, and the least squares method. Includes data for hands-on practice.

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