We are tasked with finding the equation of the least-squares regression line for predicting the etching rate $\hat{y}$ from the room temperature $x$. Given the statistics:

• Mean of room temperature $\bar{x} = 20.9$
• Standard deviation of room temperature $s_x = 3$
• Mean of etching rate $\bar{y} = 6.6$
• Standard deviation of etching rate $s_y = 0.7$
• Correlation coefficient $r = 0.99$

The formula for the least-squares regression line is:

$\hat{y} = b_0 + b_1x$

Where:

• $b_1$ (the slope) is calculated by: $b_1 = r \times \frac{s_y}{s_x}$
• $b_0$ (the y-intercept) is calculated by: $b_0 = \bar{y} - b_1 \bar{x}$

Step 1: Calculate the slope $b_1$

$b_1 = 0.99 \times \frac{0.7}{3} = 0.99 \times 0.2333 \approx 0.23$

Step 2: Calculate the intercept $b_0$

$b_0 = 6.6 - (0.23 \times 20.9) = 6.6 - 4.807 \approx 1.79$

Final equation:

$\hat{y} = 1.79 + 0.23x$

This is the equation of the least-squares regression line.

Would you like more details or have any questions?

Here are five related questions for further understanding:

1. How does the correlation coefficient $r$ affect the slope of the regression line?
2. What does the slope of 0.23 represent in the context of this problem?
3. How would you interpret the y-intercept of 1.79 in this scenario?
4. What is the significance of a correlation coefficient close to 1, such as 0.99?
5. How would predictions change if the correlation were weaker?

Tip: A high correlation value (close to 1) indicates that the linear relationship between variables is strong, making the regression line more reliable for predictions.