The New England Journal of Medicine (Li et al., 2012) published a study investigating the effects of tai chi on postural stability in patients with Parkinson’s disease. Parkinson’s disease causes impaired balance, which leads to diminished functional ability and increased risk of falling. The researchers compared three different exercise programs: tai chi, resistance training, and stretching. One-hundred ninety five, 40- to 85-year-old patients with Parkinson’s whose condition was stable due to medication use and who had the ability to stand unaided and walk without assistance were enrolled in the study. One-third (65 patients) were randomly assigned to each exercise program for 24 weeks. Changes in various outcomes from before to after were recorded. The outcome we will focus on here is the change in functional reach (measured in centimeters) from the beginning to the end of the study (a 6-month period). Functional reach is assessed as the maximal distance a participant could reach forward beyond arm’s length while standing—larger scores are better. These results are in the file Functional Reach. 

a. Identify the explanatory and response variables and describe the type (categorical or quantitative) of each. 

b. State the hypotheses in words or symbols. 

c. Put the data into the Multiple Means applet and give the three mean number changes in functional reach. Which treatment had the highest change? Which had the lowest? 

d. Do the conditions appear to be met for a theory-based ANOVA test? Explain. 

e. Use theory-based inference to find a p-value and determine whether there is strong evidence of an association between the treatment and the change in a person’s functional reach. 

f. If you have significant results, use follow-up confidence intervals to find which groups differ significantly from which other groups. g. Use the ANOVA table to help you calculate the value of R² for this study. What does that value mean in the context of the study?