Answer:y = mx + b (where y = height, x = shoe size, m = slope, b = y-intercept)
Explanation:Part A: Finding the Least-Squares Regression Line
Calculate the mean: Find the average shoe size (independent variable) and average height (dependent variable).
Calculate deviations from the mean: Subtract the mean shoe size from each shoe size value and subtract the mean height from each height value.
Calculate the product of deviations: Multiply the corresponding deviations from step 2 (shoe size and height).
Calculate the squared deviations: Square each deviation from step 2 (both shoe size and height).
Sum the products and squared deviations: Sum the products of deviations (step 3) and sum the squared deviations of both shoe size and height (step 4).
Find the slope (m): The slope represents the change in height per unit change in shoe size. Divide the sum of products (step 5) by the squared deviations of shoe size (step 4).
Find the y-intercept (b): The y-intercept represents the average height when the shoe size is zero (which is unrealistic but helps with the equation). Subtract the product of the slope (m) from step 6 and the mean shoe size (step 1) from the mean height (step 1).
Formula:
y = mx + b (where y = height, x = shoe size, m = slope, b = y-intercept)
Note: You'll need the actual data points to perform these calculations.
Part B: Finding the Residual for Carlos
Use the regression equation: Plug Carlos's shoe size (9) into the equation you obtained in Part A.
Predicted height: Calculate the predicted height based on the equation.
Residual: Subtract Carlos's actual height (71 inches) from the predicted height.
Residual = Actual Height - Predicted Height
Part C: Interpreting the Residual
The residual represents the difference between Carlos's actual height and the height predicted by the regression line. A positive residual means Carlos is taller than the average height predicted for his shoe size. Conversely, a negative residual indicates he's shorter than expected.
Note: Residuals help assess how well the regression line fits the data. In this case, a small residual suggests Carlos's data point is close to the regression line, while a large residual indicates a significant deviation.