A physiologist wishes to investigate the relationship between the physical characteristics of preadolescent boys and their maximum oxygen uptake (measured in milliliters of oxygen per kilogram of body weight). The data below (saved in the BOYS10 file) were collected on a random sample of 10 preadolescent boys.

a. Fit the regression model

y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ε

to the data, and give the least squares prediction equation.

b. It seems reasonable to assume that the greater a child’s weight, the greater should be the maximum oxygen uptake. Is β3, the estimated coefficient of weight x3, positive as expected? Give an explanation for this result.

c. It would seem that the chest depth of a child should be positively correlated with lung volume and hence to maximum oxygen uptake. Is β4 significantly different from 0, as expected? If not, explain why.

d. Calculate the correlation coefficients between all pairs of the independent variables x1-x4. Do these correlations provide an explanation for the confusing signs and small t-values associated with the estimated regression coefficients of the model?