https://file.aiquickdraw.com/mathbot/file/1733127542231joq9dpe5.jpg

To solve for the regression line based on the given data, I will calculate the equation of the line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Let me calculate these values step by step:

1. Compute the slope \( m \) using the formula:
\[
m = \frac{n\sum(xy) - \sum x \sum y}{n\sum(x^2) - (\sum x)^2}
\]

2. Compute the y-intercept \( b \) using:
\[
b = \frac{\sum y - m \sum x}{n}
\]

Where \( n \) is the number of data points. Let me calculate this. pandas as pd

# Data from the image
data = {
    'x': [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13],
    'y': [26.52, 27.3, 23.28, 23.66, 21.24, 18.42, 18.5, 16.98, 14.46, 14.64, 12.72, 9.5]
}

# Convert data to a DataFrame
df = pd.DataFrame(data)

# Calculate sums needed for the regression line calculation
n = len(df)
sum_x = df['x'].sum()
sum_y = df['y'].sum()
sum_xy = (df['x'] * df['y']).sum()
sum_x2 = (df['x']**2).sum()

# Calculate 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 regression line for the given data is:

\[
y = -1.53x + 30.41
\]

Would you like to see the detailed steps for calculating this, or have additional questions? Here are 5 related questions you might explore:

1. How is the slope calculated step by step for regression?
2. How can we interpret the slope and intercept in this context?
3. How would the regression line change if more data points were added?
4. Can you provide a plot of the data and the regression line?
5. How can we use this regression line to predict values for new \(x\)?

**Tip**: Always check the scatterplot of the data to visually verify the fit of your regression line!

Based on the data shown below, calculate the regression line (each value to two decimal places):

x = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]
y = [26.52, 27.3, 23.28, 23.66, 21.24, 18.42, 18.5, 16.98, 14.46, 14.64, 12.72, 9.5]

Calculate the linear regression line using the least squares method for the dataset (x = [2, 3, 4, ..., 13], y = [26.52, 27.3, 23.28, ..., 9.5]). This guide provides step-by-step calculations of the slope and intercept.

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