Can the right diet help us cope with diseases associated with aging such as Alzheimer’s disease? A study (“Reversals of Age-Related Declines in Neuronal Signal Transduction, Cognitive, and Motor Behavioral Deficits with Blueberry, Spinach, or Strawberry Dietary Supplement,” J. Neurosci. 1999: 8114–8121) investigated the effects of fruit and vegetable supplements in the diet of rats. The rats were 19 months old, which is aged by rat standards. The 40 rats were randomly assigned to four diets, of which we will consider just the blueberry diet and the control diet here. After 8 weeks on their diets, the rats were given a number of tests. We give the data for just one of the tests, which measured how many seconds they could walk on a rod. Here are the times for the ten control rats (C) and ten blueberry rats (B):

The objective is to obtain a 95% confidence interval for the difference of population means.
a. Determine a 95% confidence interval for the difference of population means using the two-sample t interval.
b. Use software to generate a bootstrap sample of differences of means. Check game the Rockies won 10 to 7 on a home run with two out in the bottom of the tenth inning, so there were 59 outs instead of 54, and the number of runs is adjusted to (54/59)(17) = 15.56. We want to compare the average runs in Denver with the average runs in Phoenix.
a. Find a 95% confidence interval for the difference of population means using the two-sample t interval.
b. Use software to generate a bootstrap sample of differences of means. Check the bootstrap distribution for normality using a normal probability plot.
c. Use the standard deviation of the bootstrap distribution along with the mean and t critical value from (a) to get a 95% confidence interval for the difference of means.
d. Use the bootstrap sample and the percentile method to obtain a 95% confidence interval for the difference of means.
e. Compare your three confidence intervals. If you used a standard normal critical value in place of the t critical value in (c), why would that make this interval more like the one in (d)? Why should the three intervals be fairly similar for this data set?
f. Interpret your results. Is there a substantial difference between the two locations? Compare the difference with what you thought it would be. If you were a major league pitcher, would you want to be traded to the Rockies?

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C: B: 15.00 6.24 5.12 7.91 7.00 4.12 9.38 7.38 2.44 8.21 18.77 15.09 5.60 3.90 15.03 11.57 3.63 0.95 6.67 8.98