Some studies have suggested that aerobic activity can benefit brain development and cognition in growing children (Chaddock-Heyman, 2015). It is thought that the higher your aerobic activity as measured by VO₂ max (mL/kg/min), the thinner the cortical grey matter in the superior and anterior frontal lobes (measured with an MRI). A sample of 40 children (ages 8–10 years) was recruited from Illinois to participate in the study. Use the data found in VO2MaxSFLobe to answer the questions. 

a. Use the Corr/Regression applet to look at a scatterplot of the data where VO₂ max is the explanatory variable and superior-frontal (the MRI-measured thickness of the superior frontal cortical grey matter) is the response variable. Is the association between these two variables positive or negative? What does that mean in terms of VO₂ max and thickness of cortical grey matter in the superior frontal lobe? 

b. What is the least squares regression equation? 

c. Look at the scatterplot with the least squares regression equation drawn on the plot. Are all validity conditions met to carry out a theory-based test of significance for slope? Explain. 

d. State the null and alternative hypotheses to test whether there is a linear association between VO₂ max and thickness of grey matter in the superior frontal lobe of the brain. 

e. Based on your answer to part (c), carry out either a simulation-based test or theory-based test for slope. Report your p-value. 

f. State your conclusions in terms of strength of evidence against the null based on your p-value. Also include statements of causation and generalization.