Let's go through each part of the problem shown in the image systematically. We have a dataset and calculations related to the least-squares regression line. Here’s how to solve each question:

Part (a)

The total variation in the sample $y$-values is given by the total sum of squares (SST), which is represented by $\sum (y_i - \bar{y})^2$. For this data, we find the value from the column sum under $(y_i - \bar{y})^2$, which is 77.8920.

Part (b)

The value $r^2$ is the proportion of the total variation in the sample $y$-values that is explained by the linear relationship between $x$ and $y$. It is calculated as: $r^2 = \frac{\text{SSReg}}{\text{SST}}$ where $\text{SSReg} = \sum (\hat{y}_i - \bar{y})^2 = 63.9627$ and $\text{SST} = 77.8920$.

$r^2 = \frac{63.9627}{77.8920} \approx 0.82$ So, $r^2$ rounded to two decimal places is 0.82.

Part (c)

The least-squares regression line minimizes the sum of the squared residuals, which is the column sum under $(y_i - \hat{y})^2$, equal to 15.9561. This is why it’s called the “best fit” line.

Summary Answers

1. Part (a): Total variation = $(y_i - \bar{y})^2$, value: 77.8920
2. Part (b): $r^2 = 0.82$
3. Part (c): The minimized quantity is $(y_i - \hat{y})^2$, value: 15.9561

Would you like further details or explanations on any of these points?

Relative Questions:

1. How is the least-squares regression line derived mathematically?
2. What is the significance of $r^2$ in interpreting the fit of a regression model?
3. Why is minimizing the sum of squared residuals important in regression analysis?
4. How can you determine if a regression model is appropriate for the given data?
5. How does changing the slope or intercept of the line affect the $r^2$ value?

Tip:

Always verify the sums and computations when working with least-squares regression, as small errors in the sums can lead to incorrect conclusions about the model's accuracy.