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6. [0.44/1 Points] DETAILS PREVIOUS ANSWERS BBUNDERSTAT12 11.4.007.MI. M Many economists and financial experts claim that the price level of a stock or bond is not random, but the price changes tend to follow a random sequence over time. The following data represent annual percentage returns on a stock index for a sequence of recent years. 12.2 10.8 -0.4 35.2 21.8 31.8 23. 23.2 -10.4 -11.2 -21.6 10.2 Test the sequence for randomness about the median. Use a = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. O Ho: The numbers are randomly mixed about the median. Hy: The numbers are not randomly mixed about the median. O Ho: The numbers are randomly mixed about the mean. Hy: The numbers are not randomly mixed about the mean. O Ho: The numbers are not randomly mixed about the mean. H: The numbers are randomly mixed about the mean. O Ho: The numbers are not randomly mixed about the median. H: The numbers are randomly mixed about the median. (b) Find the sample test statistic R, the number of runs. R = (c) Find the upper and lower critical values in the Critical Values for Number of Runs R table. 61 =7. [0.44/1 Points] DETAILS PREVIOUS ANSWERS BBUNDERSTAT12 11.4.010. MY NOTES ASK YOUR TEA Sand and clay studies were conducted at the West Side Field Station of the University of California. (Reference: Professor D. R. Nielsen, University of California, Davis.) Twelve consecutive depths, each about 15 cm deep, were studied and the following percentages of clay in the soil were recorded. 47.9 43.7 48.5 42.5 41.6 40.6 46.5 44.1 36.8 35.9 33.3 42.5 Test the sequence for randomness about the median. Use a = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. O H : The numbers are randomly mixed about the mean. Hy: The numbers are not randomly mixed about the mean. O H : The numbers are randomly mixed about the median. H The numbers are not randomly mixed about the median. O H : The numbers are not randomly mixed about the mean. Hy: The numbers are randomly mixed about the mean. O H: The numbers are not randomly mixed about the median. Hy: The numbers are randomly mixed about the median. (b) Find the sample test statistic R, the number of runs. (c) Find the upper and lower critical values in Table 10 of Appendix II.8. [0.37/1 Points] DETAILS PREVIOUS ANSWERS BBUNDERSTAT12 11.4.011. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER For each successive presidential term from Franklin Pierce (the 14th president, elected in 1853) to William J. Clinton, second term, (42nd president), the party affiliation controlling the White House is shown below, where R designates Republican and D designates Democrat. (Reference: The New York Times Almanac.) Historical Note: We start this sequence with the 14th president because earlier presidents belonged to political parties such as the Federalist or Wigg party (not Democratic or Republican). In cases in which a president died in office or resigned, the period during which the vice president finished the term is not counted as a new term. The one exception is the case in which Lincoln (a Republican) was assassinated and the vice president Johnson (a Democrat) finished the term. DF D DRR DR RRRDDR Test the sequence for randomness at the 5% level of significance. Use the following outline. (a) State the null and alternate hypotheses. O Ho: The symbols are not randomly mixed. Hy: The symbols are not randomly mixed. Ho: The symbols are randomly mixed. H: The symbols are randomly mixed. O Ho: The symbols are not randomly mixed. Hy: The symbols are randomly mixed. Ho: The symbols are randomly mixed. H,: The symbols are not randomly mixed. (b) Find the number of runs R, nj. n2. and the critical values, , and C2. Let n, = number of Republicans and n2 = number of Democrats. (Enter NONE in any unused answer blanks.) (c) In the case, where n, > 20, we cannot use Table 10 of Appendix II to find the critical values. Whenever either nj or n2 exceeds 20, the number of runs R has a distribution that is approximately normal, as follows. HR= - 2nin2 + 1 and OR = (2nin2) (2nin2 - n1 - 72) (n + n2)- (n + n2 - 1) We convert the number of runs R to a z value, and then use the normal distribution to find the critical values. Convert the sample test statistic R to z using the following formula. (Use 2 decimal places. NOTE, if n, s 20, enter NA in the answer blank.) R - HR OR 2 = (d) The critical values of a normal distribution for a two-tailed test with level of significance a = 0.05 are -1.96 and 1.96 (see Table 5(c) of Appendix II). Reject Ho if the sample test statistic z s -1.96 or if the sample test statistic z 2 1.96. Otherwise, do not reject Ho. Sample z s -1.96 -1.96 20, we cannot use Table 10 of Appendix II to find the critical values. Whenever either n, or /2 exceeds 20, the number of runs R has a distribution that is approximately normal, as follows. (2nin2)(2nin2 - n1 - 12) HR= 2ning + 1 and OR = \\ n1 + n2 (n +n2)- (n + n2 - 1) We convert the number of runs R to a z value, and then use the normal distribution to find the critical values. Convert the sample test statistic R to z using the following formula. (Use 2 decimal places.) R - PR 2 =