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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?

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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.

Dmitri wants to look at the relationship between room temperature and the etching rate each time he has etched a circuit board with cupric chloride. The data show a linear pattern with the summary statistics: Mean room temperature = 20.9°C, SD of room temperature = 3, Mean etching rate = 6.6 micrometers/min, SD of etching rate = 0.7 micrometers/min, Correlation coefficient r = 0.99.

Calculate the least-squares regression line for predicting the etching rate from room temperature using statistics including mean, standard deviation, and correlation coefficient. The problem explores key concepts in statistics and regression analysis, suitable for students in Grades 10-12.

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