https://file.aiquickdraw.com/mathbot/file/1733526446066dywgob9f.jpg

To assist you, I'll extract the mathematical content from the image and guide you step-by-step.

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### 1. Observations:
From the table in the image:
- \( x = \{2, 6, 9, 13, 20\} \)
- \( y = \{7, 18, 9, 25, 23\} \)

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### 2. Subtasks:
**(a)** Identify the scatterplot that represents the data.  
I'll compare the data points with the scatterplots labeled A, B, C, and D.  

**(b)** Develop the regression equation \( y = a + bx \) to 1 decimal place.  
We will compute:
1. \( b = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}} \)
2. \( a = \bar{y} - b\bar{x} \)

**(c)** Use the regression equation to predict \( y \) when \( x = 6 \).

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### Let me perform the calculations now. numpy as np

# Data points
x = np.array([2, 6, 9, 13, 20])
y = np.array([7, 18, 9, 25, 23])

# Number of points
n = len(x)

# Mean of x and y
x_mean = np.mean(x)
y_mean = np.mean(y)

# Calculating the slope (b)
b_numerator = np.sum((x - x_mean) * (y - y_mean))
b_denominator = np.sum((x - x_mean) ** 2)
b = b_numerator / b_denominator

# Calculating the intercept (a)
a = y_mean - b * x_mean

# Regression equation: y = a + bx (rounded to 1 decimal place)
a_rounded = round(a, 1)
b_rounded = round(b, 1)

# Prediction for x = 6
x_pred = 6
y_pred = a + b * x_pred
y_pred_rounded = round(y_pred, 1)

a_rounded, b_rounded, y_pred_rounded### Results:
1. **Regression Equation**:  
   The estimated regression equation is:
   \[
   y = 7.6 + 0.9x
   \]

2. **Prediction for \( x = 6 \)**:  
   Using the regression equation, the predicted value of \( y \) when \( x = 6 \) is:
   \[
   y = 12.9
   \]

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### Scatterplot:
Now I'll analyze the scatterplots to identify the one that matches the data points. Let me process this visually and confirm.### Matching Scatterplot:

By comparing the table of points with the scatterplots:
- The data points \((x, y) = (2, 7), (6, 18), (9, 9), (13, 25), (20, 23)\) align with **Scatterplot C**.  
Scatterplot C accurately represents the data.

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### Summary:
1. **Scatterplot**: **C**.
2. **Regression Equation**: \( y = 7.6 + 0.9x \).
3. **Prediction**: \( y = 12.9 \) when \( x = 6 \).

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Would you like detailed steps for the regression calculations or have additional questions?  
Here are related questions to explore:

1. How do we derive the slope formula in regression?
2. Why do outliers affect regression results?
3. Can you explain the interpretation of slope and intercept in this context?
4. What methods can validate the regression model?
5. What are limitations of linear regression for this dataset?

**Tip:** Always check scatterplots for clear trends before applying regression analysis.

Given the table of data points for x and y, determine the regression equation y = a + bx, and predict the value of y for x = 6. Additionally, identify the matching scatterplot from the options.

Learn how to perform linear regression analysis with a step-by-step guide. This example includes finding the regression equation y = 7.6 + 0.9x, predicting y for x = 6, and identifying the matching scatterplot C for the given data points. Suitable for students in Grades 9-12.

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