6. The signed-rank statistic can be represented as S

W1W2. . .Wn, where Wii if the sign of the xi  m0 with the ith largest absolute magnitude is positive (in which case i is included in S) and Wi 

0 if this value is negative (i  1, 2, 3, . . . , n).

Furthermore, when H0 is true, the Wi s are independent and P(W  i)  P(W  0)  .5.

a. Use these facts to obtain the mean and variance of S when H0 is true. Hint: The sum of the rst n positive integers is n(n  1)/2, and the sum of the squares of the rst n positive integers is n(n  1)(2n  1)/6.

b. The Wi s are not identically distributed (e.g., possible values of W2 are 2 and 0 whereas possible values of W5 are 5 and 0), so our Central Limit Theorem for identically distributed and independent variables cannot be used here when n is large. However, a more general CLT can be used to assert that when H0 is true and n 20, S has approximately a normal distribution with mean and variance obtained in (a). Use this to propose a large-sample standardized signed-rank test statistic and then an appropriate rejection region with level a for each of the three commonly encountered alternative hypotheses. Note: When there are ties in the absolute magnitudes, it is still correct to standardize S by subtracting the mean from (a), but there is a correction for the variance which can be found in books on nonparametric statistics.

c. A particular type of steel beam has been designed to have a compressive strength (lb/in2) of at least 50,000. An experimenter obtained a random sample of 25 beams and determined the strength of each one, resulting in the following data (expressed as deviations from 50,000):

10 27 36 55 73 77 81 90 95 99 113 127 129 136

150 155 159 165 178 183 192

199 212 217 229 Carry out a test using a signi cance level of approximately

.01 to see if there is strong evidence that the design condition has been violated.