Question

Use the dataset BodyFat, which gives the percent of weight made up of body fat for 100 men as well as other variables such as Age, Weight (in pounds), Height (in inches), and circumference (in \(\mathrm{cm}\) ) measurements for the Neck, Chest, Abdomen, Ankle, Biceps, and Wrist. \(^{78}\) The regression line for predicting body fat percent using neck circumference is BodyFat \(=-47.9+1.75 \cdot\) Neck (a) What body fat percent does the line predict for a person with a neck circumference of \(35 \mathrm{~cm}\) ? Of \(40 \mathrm{~cm} ?\) (b) Interpret the slope of the line in context. (c) One of the men in the study had a neck circumference of \(38.7 \mathrm{~cm}\) and a body fat percent of \(11.3 .\) Find the residual for this man.

Short answer

For a person with a neck circumference of 35 cm, the predicted body fat percent is approximately 13.25%, and for a neck circumference of 40 cm, the predicted body fat percent is approximately 22%. The slope of 1.75 suggests that for every additional cm in neck circumference, the body fat percent increases by 1.75%. The residual for the man with a neck circumference of 38.7 cm and body fat percent of 11.3 is approximately -4.4.

Step-by-step solution

Predict Body Fat% Given Neck Circumference

To predict the body fat percent for a person with a neck circumference of 35 cm and 40 cm, substitute these values into the regression equation. So, for a neck circumference of 35 cm, Body Fat %= \(-47.9+(1.75*35)\) and for a neck circumference of 40 cm, Body Fat% = \(-47.9+(1.75*40)\)

Interpret the slope

The slope of the line represents the change in the dependent variable (Body Fat%) for each unit increase in the independent variable (Neck Circumference). In this case, the slope is 1.75, which means for each cm increase in neck circumference, the body fat percent is predicted to increase by 1.75%.

Calculate the residual

To find the residual for the man with a neck circumference of 38.7 cm and a body fat percent of 11.3, first predict the body fat percent using the regression line as Body Fat% = \(-47.9 + (1.75 * 38.7)\). Now, subtract this predicted value from the actual value of body fat percent. The residual = Observed body fat% - Predicted body fat%