To solve the questions (a), (b), and (c) step by step from the provided table and regression information, let’s break it down:

Given Information:

• Regression line equation: $y' = 114.75 - 0.74x$
• Sample data: $x$ and $y$ values with their squared deviations in the table.
• Summed values:
  • $\Sigma (y - \bar{y})^2 = 204.0320$
  • $\Sigma (y' - \bar{y})^2 = 33.3407$
  • $\Sigma (y - y')^2 = 169.5501$

(a) Minimization in the least-squares regression line:

The least-squares regression line minimizes the sum of the squared residuals. Residuals are the differences between observed values $y$ and predicted values $y'$.

Correct answer for (a):

• First dropdown: "sum of squared errors"
• Second dropdown: "Σ(y - y')²"

(b) Variation not explained by the regression:

The variation not explained by the regression line is quantified as the residual sum of squares $\Sigma (y - y')^2$.

From the table:

• $\Sigma (y - y')^2 = 169.5501$

Correct answer for (b):

• First dropdown: "residual sum of squares"
• Second dropdown: "169.5501"

(c) Proportion of variation explained (R²):

The proportion of the total variation in $y$-values that can be explained by the regression line is the coefficient of determination $R^2$.

The formula for $R^2$ is:

$R^2 = \frac{\text{Regression Sum of Squares (SSR)}}{\text{Total Sum of Squares (SST)}} = \frac{\Sigma (y' - \bar{y})^2}{\Sigma (y - \bar{y})^2}$

Here:

• $\Sigma (y' - \bar{y})^2 = 33.3407$ (explained variation)
• $\Sigma (y - \bar{y})^2 = 204.0320$ (total variation)

Substitute these values:

$R^2 = \frac{33.3407}{204.0320} \approx 0.16$

Rounded to two decimal places:

Correct answer for (c): 0.16

Final Answers Summary:

(a)

• First dropdown: "sum of squared errors"
• Second dropdown: "Σ(y - y')²"

(b)

• First dropdown: "residual sum of squares"
• Second dropdown: "169.5501"

(c)

• $R^2 = 0.16$

Let me know if you want me to explain these steps further or provide a detailed breakdown! Here are 5 related questions you might want to explore:

1. What is the formula for the least-squares regression line?
2. How do you calculate the total sum of squares (SST)?
3. What is the difference between residuals and predicted values?
4. How can $R^2$ be interpreted in regression analysis?
5. What does it mean if $R^2$ is close to 1 or close to 0?

Tip: Always double-check the residuals and total sums when solving regression problems to avoid calculation errors!