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A prediction equation, like the one given in the exercise, is a mathematical formula used to estimate an outcome based on one or more predictor variables. In this exercise, the prediction equation is of the form:

• \( \hat{y} = -121 + 3.50x_1 + 1.35x_2 \)

Here, \( \hat{y} \) represents the predicted total body weight of a female athlete. The values \( x_1 \) and \( x_2 \) are the predictor variables, where \( x_1 \) is the athlete's height in inches, and \( x_2 \) is the percent body fat. The coefficients 3.50 and 1.35 represent how much weight changes with each unit change in height and body fat, respectively. The constant term, 21, adjusts the overall equation to fit the data better.
To use this equation, you substitute actual values for \( x_1 \) and \( x_2 \) and solve for \( \hat{y} \). This predicted value helps in understanding trends and estimating unknown outcomes.

Residual analysis is a crucial step in linear regression as it helps measure the accuracy of the prediction equation. A residual is the difference between the observed (or actual) value and the predicted value derived from the model. Mathematically, it is represented as:

• Residual = Actual Value - Predicted Value (\( y - \hat{y} \))

In the exercise, after predicting the total body weight using the given prediction equation, the residual is calculated for a specific athlete with actual data:

• \( \hat{y} = 134.3 \)
• Actual weight \( y = 115 \)
• Residual = \( 115 - 134.3 = -19.3 \)

This negative residual of -19.3 indicates that the model overestimated this athlete's weight by 19.3 pounds.
Residuals are important because they signal the potential issues in the model's fit, where large residuals suggest the model could be improved. It may also indicate variability in actual measurements, or areas where the model doesn't capture some aspect of the data well.

In linear regression exercises like this, multivariate analysis plays a key role. It refers to the statistical examination of more than two variables to understand their effect on an outcome. This approach provides a broader perspective than analyzing single variables independently.
The given exercise is a classic case of multivariate regression because it involves two independent variables:\( x_1 \) and \( x_2 \). By incorporating multiple predictors, analyses can provide more accurate predictions and deeper insights into how different factors (like height and body fat percentage) interact to affect body weight.

• \( x_1 \) and \( x_2 \) together affect \( \hat{y} \) in a way that no single predictor variable could on its own.
• The interaction between predictors can help identify more comprehensive strategies for managing body weight.

Using multivariate analysis allows for a more refined understanding of relationships within data. It enhances the predictive power of models and improves the ability to make informed decisions based on several interconnected aspects.