Question

A company maintains two offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of Dollars) is as follows:

Office	1	1	2	2
Employee	1	2	3	4
Salary	55.6	43.2	65.2	43.2

a) Suppose two of these employees are randomly selected from among the four (without replacement). Determine the sampling distribution of the sample mean salary X.

b) Suppose one of the two offices is randomly selected. Let X1 and X2 denote the salaries of the two employees. Determine the sampling distribution of X.

c) How does E(X) from parts (a) and (b) compare to the population mean salary?

Answer 1

office	employee	salary
1	1	55.6
1	2	43.2
2	3	65.2
2	4	43.2
	total	207.2

population mean , µ =Σx/n=    207.2   /   4   =51.8

a)

Employee1	Employee2	salary1	salary2	Sample means	P(X)	x̄ * P(X)
1	2	55.6	43.2	49.4	1/6	8.233
1	3	55.6	65.2	60.4	1/6	10.067
1	4	55.6	43.2	49.4	1/6	8.233
2	3	43.2	65.2	54.2	1/6	9.033
2	4	43.2	43.2	43.2	1/6	7.200
3	4	65.2	43.2	54.2	1/6	9.033
			Total	310.8		51.800

x̄	7.200	8.233	9.033	10.067
P(X)	1/6	2/6	2/6	1/6

mean of sample means, = Σx/n=   310.8   /   6   =   51.800

c)

let x1 and x2 be salary of two employees.

office	x1	x2	average salary	P(X)
1	55.6	43.2	49.4	0.5
2	65.2	43.2	54.2	0.5

each office has equal probability of selection =1/2=0.5

x		49.4	34.9
P(X)		0.5	0.5

c)

E(Xbar) from part a is equal to mu and E(Xbar) from part b is equal to mu