T and B lymphocytes are normal components of the immune system, but in multiple sclerosis they become autoreactive and attack the central nervous system. What triggers the autoimmune process? One hypothesis is that the disease is initiated by environmental factors, especially microbial infection. However, recent work by Berer et al. (2011) on the mouse model of the disease suggests that the autoimmune process is triggered by nonpathogenic microbes living in the gut. They compared onset of autoimmune encephalomyelitis in two treatment groups of mice from a strain that carries transgenic human CD4⁺ T cells, which initiate the disease. One group (GF) was kept free of nonpathogenic gut microbes and all pathogens. The other group (SPF) was only pathogen-free and served as controls. The following data are measurements of the percentage of T cells producing the molecule, interleukin-17, in tissue samples from 16 mice in the two groups.

a. Use a graph to visualize the data. What trend is suggested? In what way do the frequency distributions violate the assumptions of the two-sample t-test ?

b. No transformations were effective, so we tested the difference between the medians of the two populations with a permutation test. Ten thousand randomizations were carried out, and the difference between the medians was calculated each time (SPF minus GF). Below we list the 100 largest values for the difference in median (sorted, out of 10,000). Using these values, complete the permutation test.

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SPF: 18.87,18.87, 15.65,15.65, 13.45,13.45, 12.95,12.95, 6.01,6.01, 5.84, 5.84, 3.56,3.56, 3.463.46 GF: 6.64,6.64, 4.51,4.51, 1.12,1.12, 0.62,0.62, 0.37,0.37, 0.61,0.61, 0.71, 0.71, 0.820.82