(b)

Find the equation of the least-squares regression line. (Round your values to three decimal places.)

ŷ = 

 + 

 
 

x

(c)

Calculate the five residuals. (Round your answers to three decimal places.)

Representative

Age (x)	Residual

4.0	

7.0	

10.0	

13.5	

17.0	


To find the equation of the least-squares regression line and calculate the residuals, we need the following:

1. **Equation of the least-squares regression line**:  
   The formula is \( \hat{y} = b_0 + b_1 x \), where:
   - \( b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \) is the slope,
   - \( b_0 = \bar{y} - b_1 \bar{x} \) is the intercept.

2. **Residuals**:  
   A residual for each data point is calculated as:
   \[
   \text{Residual} = y_i - \hat{y}_i
   \]
   where \( y_i \) is the actual value and \( \hat{y}_i \) is the predicted value from the regression line.

To proceed, I'll need the corresponding \( y \)-values (dependent variable) for the given \( x \)-values: \( 4.0, 7.0, 10.0, 13.5, 17.0 \). Can you provide the \( y \)-values, or should I assume them to calculate the regression line and residuals?

Let me know if you want more details or have any specific questions! Here are some questions to deepen your understanding:

1. What is the significance of the least-squares regression line?
2. How is the slope of the regression line interpreted in real-world contexts?
3. Why do we calculate residuals in regression analysis?
4. What happens when residuals are large?
5. How can the residual plot help in diagnosing model fit?

**Tip**: Always visualize the regression line and residuals to get better insights into the model's accuracy!

Find the equation of the least-squares regression line. (Round your values to three decimal places.) Calculate the five residuals. (Round your answers to three decimal places.)

Learn how to find the equation of the least-squares regression line and calculate residuals for a set of data points. This problem covers statistical concepts such as regression lines, residuals, and how to compute them using key formulas. Suitable for high school students in Grades 10-12.

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