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To calculate the value of \( \beta_1 \) (the slope) of the regression equation using the provided data, we'll use the formula for the slope in simple linear regression:

\[
\beta_1 = \frac{n \sum (xy) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}
\]

Where:
- \( x \) and \( y \) are the values of the independent and dependent variables, respectively.
- \( n \) is the number of data points.

Let's go through the calculation step-by-step:

1. **Calculate the sums needed:**
   - \( \sum x \)
   - \( \sum y \)
   - \( \sum xy \)
   - \( \sum x^2 \)

2. **Plug in the values and compute \( \beta_1 \).**

Let's calculate these values.The value of \( \beta_1 \) (the slope of the regression line) is approximately **3.261**. This means that for every unit increase in the amount of fertilizer applied (x), the yield of potatoes (y) increases by approximately 3.261 kg, on average.

Do you have any questions, or would you like more details?  
Here are some related questions for further exploration:

1. What is the intercept (\(\beta_0\)) of the regression equation?
2. How can we interpret the slope in the context of this problem?
3. What are the assumptions behind the linear regression model?
4. How can we assess the goodness of fit for this regression model?
5. What is the significance of the residuals in a regression analysis?
6. How can outliers affect the results of a linear regression?
7. What are some methods to visualize the relationship between x and y?
8. How would you compute the predicted yield for a specific amount of fertilizer?

**Tip:** When interpreting the slope of a regression line, always consider the context of the variables involved to understand the real-world implications of the relationship.

Learn how to calculate the slope (\( \beta_1 \)) of a regression line using the formula for simple linear regression. Step-by-step guide with example calculations provided. Ideal for students and professionals interested in statistics and data analysis.

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