19 Case A: Body fat in women

Do thigh and midarm both have no effect on body fat when triceps is in the model? Do the relationships among the predictors cause any problems in the fitted model?

It is expensive and cumbersome to determine the body fat in humans as it involves immersion of the person in water. This dataset provides information on body fat, triceps skinfold thickness, thigh circumference, and mid-arm circumference for \(20\) healthy females aged \(20\) to \(34\). It would therefore be helpful if a regression model with some or all of these predictor variables could provide reliable predictions of the amount of body fat, since the measurements needed for the predictor variables are easy to obtain.

The data file is BodyFat.xlsx. The variables are:

Body Fat: Data Labels
Label Description
Triceps Triceps skinfold thickness
Thigh Thigh circumference
Midarm Midarm circumference
BodyFat Body fat


Block I:

  1. Perform a detailed regression analysis on Model 1.
  2. Is the relationship between dependent and independent variables linear?
  3. Interpret the slope.
  4. Test if the slope is significantly different from 0.
  5. Predict the dependent variable for Triceps = 30.9 , 18.2 and 14.5
  6. Interpret the \(R^2.\)
  7. Fit and interpret the model when log transformation is used for both variables.

Block II:

  1. Analyze the dependent variable. Is it normally distributed?
  2. Analyze the independent variables and describe its relationship with Body Fat.
  3. Fit and interpret Models 1, 2 and 3. Which one is the best predictor for Body Fat? Why?
  4. Analyze the relationship among the independent variables.
  5. Fit and interpret Model 4. Is this model useful to estimate Body Fat? Why?
  6. Test whether Thigh and Midarm both have no effect on Body Fat when Triceps is in the model
  7. Based on the previous results. What model do you recommend? Elaborate your answer.


  • Model 1: \[\text{BodyFat}_i=\beta_0+\beta_1\text{Triceps}_i+\epsilon_i\]

  • Model 2: \[\text{BodyFat}_i=\beta_0+\beta_1\text{Thigh}_i+\epsilon_i\]

  • Model 3: \[\text{BodyFat}_i=\beta_0+\beta_1\text{Midarm}_i+\epsilon_i\]

  • Model 4: \[\text{BodyFat}_i=\beta_0+\beta_1\text{Triceps}_i+\beta_2\text{Thigh}_i+\beta_3\text{Midarm}_i+\epsilon_i\]


References: