Question

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

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

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

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.

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.

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.