To solve this regression problem, I need to calculate the equation of the regression line for the given data points. Here’s the breakdown of how to approach it:

Steps to Solve:

1. Find the regression line using all 10 points:

   • Calculate the mean of the x-values ($\bar{x}$) and y-values ($\bar{y}$).
   • Compute the slope ($m$) using the formula: $m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
   • Calculate the y-intercept ($b$) using the equation: $b = \bar{y} - m\bar{x}$
   • Write the regression equation: $\hat{y} = b + mx$.

2. Remove the outlier at (2,3) and repeat the calculations:

   • Recompute the regression line without the point (2,3) by recalculating $\bar{x}$, $\bar{y}$, $m$, and $b$.

3. Compare the results of both equations (with and without the outlier).

   • Assess how much the regression line changed due to the removal of the outlier.

Let me calculate the regression equations for you using this process. as np import pandas as pd

Data points from the graph

data_all_points = { "x": [2, 4, 4, 4, 5, 5, 5, 6, 6, 6], "y": [3, 7, 8, 9, 6, 7, 8, 6, 7, 8] }

Remove the outlier point (2, 3)

data_without_outlier = { "x": [4, 4, 4, 5, 5, 5, 6, 6, 6], "y": [7, 8, 9, 6, 7, 8, 6, 7, 8] }

Convert to DataFrame for calculation

df_all = pd.DataFrame(data_all_points) df_no_outlier = pd.DataFrame(data_without_outlier)

Function to calculate regression line

def regression_line(df): x_mean = df["x"].mean() y_mean = df["y"].mean() slope = ((df["x"] - x_mean) * (df["y"] - y_mean)).sum() / ((df["x"] - x_mean) ** 2).sum() intercept = y_mean - slope * x_mean return round(intercept, 3), round(slope, 3)

Calculate regression line for both datasets

intercept_all, slope_all = regression_line(df_all) intercept_no_outlier, slope_no_outlier = regression_line(df_no_outlier)

Results

results = { "All Points": {"Intercept": intercept_all, "Slope": slope_all}, "Without Outlier": {"Intercept": intercept_no_outlier,### Regression Line Results:

1. Using all 10 points:

   • Equation: $\hat{y} = 4.000 + 0.617x$

2. After removing the outlier (2, 3):

   • Equation: $\hat{y} = 9.833 - 0.500x$

Comparison:

• The regression line changes significantly after removing the outlier:
  • The slope changes from positive ($0.617$) to negative ($-0.500$).
  • The intercept shifts from $4.000$ to $9.833$.
• This demonstrates the impact of outliers on linear regression models.

Would you like a detailed explanation of any step or additional visualization of the results?

Related Questions:

1. What is the impact of outliers on the slope and intercept of regression lines?
2. How is the regression formula derived mathematically?
3. Why does the removal of a single data point (outlier) significantly alter the results?
4. What methods can be used to identify and handle outliers in a dataset?
5. How can residual plots help in understanding the fit of a regression model?

Tip:

Always visualize your data before applying regression to identify potential outliers or unusual patterns.