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. 1 0.3 2 0.4 3 0.1 4 0.2 p(x) (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 3.5 4 POCO (b) Refer to part (a) and calculate P < 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).] R 0 P(R) (d) If a random sample of size n = 4 is selected, what is PĂ S 1.5)? [Hint: You should not have to list all possible outcomes, only those for which s 1.5.]

Answer 1

below is sampling distribution :

x1	x2	probabilityP(x1,x2)		x̅	Range:
1	1	0.09		1	0
1	2	0.12		1.5	1
1	3	0.03		2	2
1	4	0.06		2.5	3
2	1	0.12		1.5	1
2	2	0.16		2	0
2	3	0.04		2.5	1
2	4	0.08		3	2
3	1	0.03		2	2
3	2	0.04		2.5	1
3	3	0.01		3	0
3	4	0.02		3.5	1
4	1	0.06		2.5	3
4	2	0.08		3	2
4	3	0.02		3.5	1
4	4	0.04		4	0

1)

xbar	1	1.5	2	2.5	3	3.5	4
p(xbar)	0.09	0.24	0.22	0.2	0.17	0.04	0.04

b)

P(Xbar<=2.5)=0.09+0.24+0.22+0.2 =0.75

c)

R	0	1	2	3
P(R )	0.3	0.36	0.22	0.12

d)

P(x̅ <=1.5)=P(1,1,1,1)+4*P(1,1,1,2)+6*P(1,1,2,2)+4*P(1,1,1,3)=0.3^4+4*0.3^3*0.4+6*0.3^2*0.4^2+4*0.3^3*0.1=

0.1485