To calculate the equation of the regression line ($y = mx + b$), we need the slope ($m$) and the y-intercept ($b$). The formula for these are:

1. Slope ($m$): $m = r \cdot \frac{s_y}{s_x}$ where:

• $r$ is the correlation coefficient,
• $s_y$ is the standard deviation of $y$,
• $s_x$ is the standard deviation of $x$.

2. Y-intercept ($b$): $b = \bar{y} - m \cdot \bar{x}$ where:

• $\bar{x}$ is the mean of $x$,
• $\bar{y}$ is the mean of $y$.

Given values:

• $r = -0.98$,
• $s_x = 7$,
• $s_y = 3$,
• $\bar{x} = 399$,
• $\bar{y} = 85$.

Step 1: Calculate $m$

$m = -0.98 \cdot \frac{3}{7} = -0.98 \cdot 0.42857 \approx -0.42$

Step 2: Calculate $b$

$b = 85 - (-0.42) \cdot 399$ $b = 85 + 167.58 = 252.58$

Regression Equation:

$y = -0.42x + 252.58$

Final Answer:

The equation of the regression line is: $y = -0.42x + 252.58$

Would you like me to verify the calculation, explain any steps in more detail, or help with further related concepts?

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Follow-Up Questions:

1. How does the correlation coefficient influence the slope of the regression line?
2. Why is the y-intercept $b$ significant in the equation of the line?
3. What does a negative slope mean in the context of this problem?
4. How would the regression line change if $r$ were closer to 0?
5. What are the assumptions behind using linear regression for a dataset?

Tip: Always visualize your data before relying on regression models to ensure linearity and detect outliers!