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Chapter 2: Problem 202

Two variables are defined, a regression equation is given, and one data point is given. (a) Find the predicted value for the data point and compute the residual. (b) Interpret the slope in context. (c) Interpret the intercept in context, and if the intercept makes no sense in this context, explain why. Weight \(=\) maximum weight capable of bench pressing (pounds), Training = number of hours spent lifting weights a week. Weight \(=95+11.7\) (Training); data point is an individual who trains 5 hours a week and can bench 150 pounds.

Short Answer

Expert verified
The predicted weight of an individual training 5 hours a week is calculated using the given regression equation, resulting in a calculated predicted weight. The residual is the difference between this and the individual's actual weight of 150 pounds. The slope of the regression equation, 11.7, indicates that for each additional hour of training per week, the individual's maximum bench press weight increases by approximately 11.7 pounds. The intercept suggests that an individual who does not train at all could still bench press 95 pounds, which may not make sense in all situations.

Step by step solution

01

Calculate Predicted Value

Using the regression equation (Weight = 95 + 11.7 * Training), and given that training is 5 hours a week, the predicted weight (Weight_predicted) should be calculated as follows: Weight_predicted = 95 + 11.7 * 5
02

Compute Residual

The residual is the difference between the actual weight and predicted weight, i.e., Residual = Actual Weight - Predicted Weight. Given the actual Weight is 150 pounds, plug it into the equation to get the residual.
03

Interpret the Slope

The slope in the regression equation is 11.7. This represents the change in the dependent variable (Weight) for each unit change in the independent variable (Training). Therefore, in this context, it suggests that for each additional hour spent lifting weights a week, the maximum weight capable of bench pressing increases by approximately 11.7 pounds.
04

Interpret the Intercept

The intercept in the regression equation is 95. This represents the predicted value of the dependent variable (Weight) when the independent variable (Training) is zero. In this context, it suggests that someone who does not spend any time weight lifting in a week is still capable of bench pressing 95 pounds. If it does not make sense, it could be because even without training, different individuals have different base strength levels, and it's impossible for everyone to have the same base strength of 95 pounds.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Residual Calculation
Residual calculation is an integral part of regression analysis. It helps us understand how well our model predicts actual data.
The residual is the difference between the observed value and the predicted value obtained from our regression model.
In simpler terms, it tells us how far off we are from the actual data point.

To compute the residual, you would follow these steps:
  • Calculate the predicted value using the regression equation. In this exercise, with a training input of 5 hours, the predicted weight is calculated as \( 95 + 11.7 \times 5 = 153.5 \) pounds.
  • Subtract the predicted value from the actual observed value to get the residual. Here, the actual weight is 150 pounds, so the residual is \( 150 - 153.5 = -3.5 \).
A negative residual implies the predicted value was overestimated, and a positive residual indicates an underestimation.
In our example, the negative residual of -3.5 suggests that the actual bench press weight was 3.5 pounds less than what our model predicted.
Slope Interpretation
The slope is an important feature of the regression equation. It tells us how the dependent variable is expected to change as the independent variable changes. In this context, the slope has a value of 11.7.

This number indicates that for every one-hour increase in training per week, the maximum bench press weight is expected to increase by approximately 11.7 pounds.
In a real-world scenario, this helps to quantify the benefit of an additional hour of training.
  • The slope's positive nature conveys that more training correlates with greater bench press strength.
  • It allows us to make informed predictions about future outcomes, assuming the linear relationship holds consistent.
Remember, the slope is a powerful tool in understanding relationships between variables. It provides insights to make strategic decisions.
Intercept Interpretation
The intercept in a regression equation is the value of the dependent variable when the independent variable is zero. In this exercise, our intercept is 95.

This suggests that if one spends zero hours a week training, they are predicted to be capable of bench pressing 95 pounds.
While this number might seem arbitrary, it does serve a purpose:
  • It provides a starting point for the regression equation, helping define where our line crosses the Y-axis.
  • It attempts to reflect some inherent level of base strength without training.
However, the intercept may not always be meaningful. Human variability in base strength means not everyone can bench press 95 pounds without training.
It鈥檚 essential to interpret the intercept within the context of your data and remember it might not always make perfect sense in every situation.

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