To produce the regression line, we need to calculate the slope ($b$) and the intercept ($a$) for the regression equation:

$\hat{y} = a + b x$

Step 1: Calculate the Slope ($b$)

The formula for the slope $b$ is: $b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$

Using the values given: $\sum (x_i - \bar{x})(y_i - \bar{y}) = 153.6$ $\sum (x_i - \bar{x})^2 = 176.4$

Substitute these values into the formula: $b = \frac{153.6}{176.4}$

Calculating: $b \approx 0.8708$

Step 2: Calculate the Intercept ($a$)

The formula for the intercept $a$ is: $a = \bar{y} - b \cdot \bar{x}$

Using: $\bar{y} = 50.5, \quad \bar{x} = 18.9$

Substitute $b$ and these values into the formula: $a = 50.5 - (0.8708 \times 18.9)$

Calculating: $a \approx 33.0379$

Final Regression Line

Now that we have both $a$ and $b$, we can write the regression line: $\hat{y} = 33.0379 + 0.8708x$

Final Answer

$\hat{y} = 33.0379 + 0.8708x$

Would you like further details on the calculations, or do you have any other questions?

Here are five related questions:

1. How is the slope of a regression line interpreted in a real-world context?
2. What does the intercept represent in a regression analysis?
3. How does the confidence level affect the interpretation of the regression line?
4. What is the purpose of calculating the sum of squares in regression?
5. How would you interpret the strength of the relationship between $x$ and $y$ here?

Tip: Always check the residuals after creating a regression line to assess the fit of the model.