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 34 p(x) | .4 .3 .2 .1

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.

b. Refer to part

(a) and calculate P(X  2.5).

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).]

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.]