Refer to Exercises 10.CE.25 and 10.CE.26 on VO₂ max and cortical grey matter in the brain. The paper’s title, “The Role of Aerobic Fitness in Cortical Thickness and Mathematics Achievement in Preadolescent Children,” indicates there might be an association between VO₂ max (a measure of aerobic fitness) and mathematics ability. The participants in the VO₂ max study also took achievement tests in mathematics, and the results are in the dataset VO2MaxMath.

a. Use the Corr/Regression applet to look at a scatterplot of the data where VO₂ max is the explanatory variable and math score is the response variable. Is the association between these two variables positive or negative? Do kids with higher VO₂ max measurements tend to have higher math scores? 

b. 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? 

c. State the null and alternative hypotheses to test whether there is a linear association between math score and VO₂ max. 

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

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

Data from Exercises 10.CE.25 

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.