Testing for Trends A turbocharger wheel is manufactured using an investment-casting process. The shaft fits into the wheel opening, and this wheel opening is a critical dimension. As wheel wax patterns are formed, the hard tool producing the wax patterns wears. This may cause growth in the wheel-opening dimension. Ten wheel-opening measurements, in time order of production, are 4.00 (millimeters), 4.02, 4.03, 4.01, 4.00, 4.03, 4.04, 4.02, 4.03, 4.03.
(a) Suppose that p is the probability that observation Xi+5 exceeds observation Xi. If there is no upward or downward trend, Xi+5 is no more or less likely to exceed Xi or lie below Xi. What is the value of p?
(b) Let V be the number of values of i for which Xi+5 Xi. If there is no upward or downward trend in the measurements, what is the probability distribution of V?
(c) Use the data above and the results of parts (a) and (b) to test H0: there is no trend, versus H1: there is upward trend. Use a = 0.05.
Note that this test is a modification of the sign test. It was developed by Cox and Stuart.