To solve this, we calculate the linear regression line $y = a + b \cdot x$, where $a$ is the y-intercept and $b$ is the slope of the line.

Step 1: Formula for slope $b$

$b = \frac{n \sum (x \cdot y) - \sum x \cdot \sum y}{n \sum x^2 - (\sum x)^2}$

Step 2: Formula for intercept $a$

$a = \frac{\sum y - b \cdot \sum x}{n}$

Step 3: Calculating required values

Let’s calculate the necessary sums for the table provided:

Totals:

$\sum x = 54, \quad \sum y = 124.7, \quad \sum x^2 = 384, \quad \sum (x \cdot y) = 844.99, \quad n = 9$

Step 4: Plug values into the formulas

Slope $b$:

$b = \frac{9 \cdot 844.99 - 54 \cdot 124.7}{9 \cdot 384 - 54^2}$ $b = \frac{7604.91 - 6733.8}{3456 - 2916}$ $b = \frac{871.11}{540} \approx 1.61$

Intercept $a$:

$a = \frac{\sum y - b \cdot \sum x}{n}$ $a = \frac{124.7 - 1.61 \cdot 54}{9}$ $a = \frac{124.7 - 86.94}{9} \approx \frac{37.76}{9} \approx 4.2$

Regression Equation:

$y = 4.2 + 1.61x$

Step 5: Estimate $y$ when $x = 13$

$y = 4.2 + 1.61 \cdot 13$ $y = 4.2 + 20.93 = 25.13$

Final Answer:

The regression equation is: $y = 4.2 + 1.61x$

When $x = 13$, $y \approx 25.13$.

Would you like a detailed explanation of any step? 😊
Here are five related questions to explore further:

1. What is the meaning of the slope and y-intercept in this context?
2. How do outliers affect the linear regression line?
3. Could a nonlinear regression model fit this data better? Why or why not?
4. What is the coefficient of determination ($R^2$) for this regression line?
5. How would you calculate this regression line in Excel or Python?

Tip: Double-check calculations when working with large datasets to avoid rounding errors!