To determine the least-squares regression line $y = a + bx$, we need to calculate the slope $b$ and the intercept $a$ of the line using the formulas:

1. Slope $b$: $b = r \cdot \frac{s_y}{s_x}$ where:

   • $r = -0.9597$ (correlation coefficient),
   • $s_y = 2.0425$ (standard deviation of $y$),
   • $s_x = 2.5100$ (standard deviation of $x$).

2. Intercept $a$: $a = \overline{y} - b \cdot \overline{x}$ where:

   • $\overline{y} = 3.9000$ (mean of $y$),
   • $\overline{x} = 3.5000$ (mean of $x$).

Let's calculate these values:

Step 1: Calculate $b$ (Slope)

$b = -0.9597 \cdot \frac{2.0425}{2.5100}$ Calculating this:

$b \approx -0.9597 \cdot 0.8137 \approx -0.7805$

Step 2: Calculate $a$ (Intercept)

$a = 3.9000 - (-0.7805) \cdot 3.5000$ Calculating this:

$a \approx 3.9000 + 2.7318 \approx 6.6318$

Final Regression Equation

Thus, the least-squares regression line is:

$y = 6.6318 - 0.7805x$

Would you like a breakdown of each calculation step or further clarification on any part?

Here are some related questions to deepen understanding:

1. How does the correlation coefficient influence the slope of the regression line?
2. What does a negative slope indicate about the relationship between $x$ and $y$?
3. How would the regression line change if the mean values of $x$ or $y$ were different?
4. How can the regression line be used to predict values of $y$ for given $x$ values?
5. What is the significance of using least squares for regression?

Tip: Remember, a high correlation coefficient (near -1 or 1) indicates a strong linear relationship between variables.