Part A 1. Let X1 be a random variable from a distribution with a mean of 8 and a standard deviation of 6 and X2 be a random variable from a distribution with a mean of 7 and a standard deviation of 5. (a) For independent samples of sizes n1 = 3 and n2 = 6, find the mean and standard deviation of Ill-X2 . (3 marks) (b) Do X1 and X2 have to be normally distributed for your answer to part (a) to be valid? Explain. (2 marks) (c) Can you conclude that the variable IllX2 in part (a) is normally distributed? Explain. (3 marks) (d) Suppose X1 and X2 are normally distributed, determine the percentage of all pairs of independent samples of sizes n1 = 3 and n2 =,6 with the property that the difference between the sample means, XlX2 . is between -9 and 11. (5 marks) 2. A study examined the sense of direction of 30 male and 30 female students. After being taken to an unfamiliar wooded park. the students were given some spatial orientation tests, including pointing south, which tested their absolute frame of reference. The students pointed by moving a pointer attached to a 3600 protractor. Following are the absolute pointing errors, in degrees. of the participants. Mlle Fen-II 13 DO 39 33 IO 8 20 3 138 13 68 18 3 ll 78 69 Ill 3 38 23 60 5 9 3} IS 35 Ill 59 5 86 22 TD 36 27 32 35 SB 3 161 IS 30 27 ll 3 lit) 8 20 61 26 I9 63 66 I76 15 (a) Given the statistical summaries, test at the 1% significance level whether males have a better sense of direction than females, on average