Question

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 P(x) 0.2 2 0.4 3 4 0.3 0.1 (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. 1.5 35 (b) Refer to part (a) and calculate PX $ 2.5). (c) Again consider a random sample of size 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).] PER) (d) If a random sample of size 4 is selected, what is PX S 1.5) (Mint: You should not have to list all possible outcomes, only those for which x $ 1.5.]

Answer 1

Solution:

(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 1 1.5 2 2.5 3.5 4 Plx) 0.0625 0.125 0.0625 0.1875 0.1875 0.25 0.125 (b) Refer to part (a) and calculate PX s 2.5) 0.625 (0.0625+0.125+0.1875+0.25=0.625) (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).]| 1 PLR) 0.25 0.375 0.25 0.125 (d) If a random sample of size n 4 is selected, what is POX 1.5)7 Hint: You should not have to list all possible outcomes, only those for which xs1.5.

Explanation:

(a)

The possible samples with their means is given below:

Samples	Sample means
1,1	1
1,2	1.5
1,3	2
1,4	2.5
2,1	1.5
2,2	2
2,3	2.5
2,4	3
3,1	2
3,2	2.5
3,3	3
3,4	3.5
4,1	2.5
4,2	3
4,3	3.5
4,4	4

Therefore, the sampling distribution of means is:

P(x) 1 = 0.0625 16 1 2 0.125 1.5 16 = 0.1875 16 4 = 0.25 16 2.5 3 = 0.1875 16 2 - 0.125 16 3.5 1 0.0625 4 16

c)

The possible samples with their ranges are:

Samples	Range
1,1	0
1,2	1
1,3	2
1,4	3
2,1	1
2,2	0
2,3	1
2,4	2
3,1	2
3,2	1
3,3	0
3,4	1
4,1	3
4,2	2
4,3	1
4,4	0

Therefore, the sampling distribution of Ranges is:

P('R) Range 4 = 0.25 16 0 6 0.375 1 16 4 - 0.25 16 2 2 = 0.125 16