A study was conducted to see whether glaucoma can be linked with another eye defect, decreased corneal thickness. Patients who had glaucoma in one eye but not the other were studied, and the following data on corneal thickness was obtained:
patient 1 2 3 4 5 6 7 8
glaucomatous eye 488 466 480 426 440 410 473 463
contralateral eye 481 481 494 424 439 420 481 466
(a) Assuming that the eight paired differences are a random sample from a normal distribution, test the hypothesis that the mean difference between the corneal thicknesses of glaucomatous and contralateral eyes is zero. Use R or a table, report your conclusion as an approximate p-value, and interpret the result. Do not use a non-parametric test.
(b) Use a non-parametric method to test the hypothesis that the difference in corneal thickness between the glaucomatous and contralateral eyes is symmetric around zero (against the alternative that glaucoma is associated with decreased corneal thickness). Calculate the p-value exactly by listing the possible outcomes with their probabilities. Do not use a table.