Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of X is as follows. x 1 2 3 4 p(x) 0.2 0.1 0.4 0.3 (a) Consider a random sample of size n = 2 (two customers), and let X be the sample mean number of packages shipped. Obtain the probability distribution of X. 4 1 1.5 2 2.5 3 3.5 P(X) 0.04 0.04 0.17 0.2 0.22 .24 0.09 (b) Refer to part (a) and calculate P(X ≤ 2.5). 0.45 (c) Again consider a random sample of size n = 2, but now focus on the statistic R = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of R. [Hint: Calculate the value of R for each outcome and use the probabilities from part (a).] R 0 P(R) 0.3 1 0.36 2 3 0.22 0.12 (d) If a random sample of size n = 4 is selected, what is P(X ≤ 1.5)? [Hint: You should not have to list all possible outcomes, only those for which x ≤ 1.5.] 0.0045