Question

A company maintains three 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	3	3
Employee	1	2	3	4	5	6
Salary	30.7	34.6	31.2	34.6	26.8	30.7

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

X.

(Enter your answers for p(x) as fractions.)

x	28.75		30.70	30.95	32.65		34.60
p(x)		1 15				2 15

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

X.

(Enter your answers as fractions.)

x	28.75	32.65	32.90
p(x)

(c) How does

E(X)

from parts (a) and (b) compare to the population mean salary μ?

E(X)

from part (a) is  ---Select--- greater than less than equal to μ, and

E(X)

from part (b) is  ---Select--- greater than less than equal to μ.

Answer 1

Solution

Part (a)

Sampling distribution of the sample mean salary X

x	Frequency	p(x)
28.75	2	2/15
29.00	1	1/15
30.70	3	3/15
30.95	2	2/15
32.65	4	4/15
32.90	2	2/15
34.60	1	1/15
Total	15	1

Answer 1

Details

Sample #	Sample	Values	Sample Mean
1	30.7	34.6	32.65
2	30.7	31.2	30.95
3	30.7	34.6	32.65
4	30.7	26.8	28.75
5	30.7	30.7	30.70
6	34.6	31.2	32.90
7	34.6	34.6	34.60
8	34.6	26.8	30.70
9	34.6	30.7	32.65
10	31.2	34.6	32.90
11	31.2	26.8	29.00
12	31.2	30.7	30.95
13	34.6	26.8	30.70
14	34.6	30.7	32.65
15	26.8	30.7	28.75

Part (b)

Sampling distribution of the sample mean salary X

x	Frequency	p(x)
28.75	1	1/3
32.65	1	1/3
32.90	1	1/3
Total	3	1.00

Answer 2

Details

Sample #	Sample	Values	Sample Mean
1	30.7	34.6	32.65
2	31.2	34.6	32.90
3	26.8	30.7	28.75

Part (c)

E(X) = Σ{x.p(x)} summed over all possible values of x…........................................................…. (1)

Vide (1),

For Part (a), E(X) = 31.43 Answer 3

Details

x	p(x)	x.p(x)
28.75	0.1333	3.8333
29.00	0.0667	1.9333
30.70	0.2000	6.1400
30.95	0.1333	4.1267
32.65	0.2667	8.7067
32.90	0.1333	4.3867
34.60	0.0667	2.3067
Total	1.00	31.4333

For Part (b), E(X) = 31.43 Answer 4

Details

x	p(x)	x.p(x)
28.75	0.3333	9.5833
32.65	0.3333	10.8833
32.90	0.3333	10.9667
Total	1.00	31.4333

Population Mean Salary μ = 31.43 Answer 5

Details

#	Value
1	30.7
2	31.2
3	26.8
4	34.6
5	34.6
6	30.7
Total	188.6
Mean	31.4333

Comparison

Vide Answers 3, 4 and 5,

E(X) from part (a) is  equal to μ Answer 6

E(X) from part (b) is  equal to μ Answer 7

DONE