Question

Sampling Distributions: To get an idea of what is going on when you are talking about a sampling distribution, consider a population of 5 test scores from a small class. The test scores are {100, 95, 90, 85, 80}. Let us obtain samples of size 2 from this population with replacement. In other words, after you select the first test score, you put it back for selection again. Sampling with replacement from a small population is similar to sampling from a large population. 1. List all 25 possible samples. For example, (80,80) and (80,85) are two of the 25 samples possible. Note that I am writing the selected scores in parentheses separated by a comma. 2. Now, give each of the 25 x bars associated with the samples above. 3. Now compute the mean of the sample means. In other words, what is mu sub x bar? 4. What is the mean of the population? 5. What is the standard deviation of the sample means from (2) above? You should use Microsoft Excel to answer this question. Please remember to calculate the population standard deviation since the 25 x bars represents all of our sample means. 6. What is the population standard deviation of the original population? Use Excel to calculate this as well. 7. Take the standard deviation found in (6) and divide by square root of 2. What do you obtain? Please comment on why this is not a surprise

Answer 1

Given S = { 100, 95, 90, 85, 80 }

There N = Population Size = 5

Sample Size = n = 2

We have to draw the samples using the technique of Simple Random Sample With Replacement.

Therefore using SRSWR we can draw

(1) and (2)

Sample No:	Samples	SAMPLE MEAN
1	(100, 100)	100
2	(100, 95)	97.5
3	(100, 90)	95
4	(100, 85)	92.5
5	(100, 80)	90
6	(95, 95)	95
7	(95, 90)	92.5
8	(95, 85)	90
9	(95, 80)	87.5
10	(95, 100)	97.5
11	(90, 90)	90
12	(90, 85)	87.5
13	(90, 80)	85
14	(90, 100)	95
15	(90, 95)	92.5
16	(85, 85)	85
17	(85, 80)	82.5
18	(85, 100)	92.5
19	(85, 95)	90
20	(85,90)	87.5
21	(80, 80)	80
22	(80, 100)	90
23	(80, 95)	87.5
24	(80, 90)	85
25	(80, 85)	82.5
	MEAN OF SAMPLE MEANS	90
	S.D OFSAMPLE MEANS	5

(3) The Mean of the Sample Means = 90

	X
	100
	95
	90
	85
	80
POPULATION MEAN	90
POPULATION S.D	7.0711

(4) The Mean of Population is also 90

(5) The S.D of the sample means = 5

(6) The Population S.D = 7.0711

(7)

which is equals to the S.D of Sample Means.