An obedience school for dogs experimented with two methods of training. One method involved rewards (food, praise); the other involved no rewards. The dogs were randomly placed into two independent groups of 11 each. The number of sessions required to train each of 22 dogs follows.

Use a 0.05 level of significance to test the hypothesis that the number of sessions was the same for the two groups against the alternate hypothesis that the number of sessions was not the same.
(a) State the test used.
(b) Give a. State the null and alternate hypotheses.
(c) Find the sample test statistic.
(d) For the sign test, rank-sum test, and Spearman correlation coefficient test, find the P-value of the sample test statistic. For the runs test of randomness, find the critical values from Table 10 of Appendix II.
(e) Conclude the test and interpret the results in the context of the application

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With rewards 12 17 15 10 16 20 9 23 8 14 10 No rewards 19 22 1 18 13 25 24 28 21 20 21