Illustrative Examples

5.2 Illustrative Examples

Imagine sampling 2 units, n = 2, from a population of size N = 7. Population size is denoted by capital N and sample size lower case n. Table ?? contains the unit labels, numbered 1 to 7, and their associated values. The population mean of this sample is μ_y = 57.00. Table ?? represents all possible samples of size n = 2 from a simple random sample with replacement. In a SRSWR, each unit has a probability of 1 - ( N--1
N )ⁿ = 1 - ( 6
7 )² = 13
49 of being in the sample. Each possible sample in Table ?? is equally likely with probability 1-
49 , because there are 49 possible samples. Table ?? illustrates SRSWR with the sampling frame being all units in the target population, or simply the population. Often when sampling some units in the target population are not in the sample frame and thus have a zero probability of being included in the sample. Imagine if a six sided die was used to select the units to be sampled. Thus the units labeled 1 to 6 would have a probability of 11
36 of being in the sample and unit number 7 has a zero probability. Table ?? illustrates the latter situation, with the sampling frame being being units labeled 1 to 6 within the population.

Table ?? represents all possible samples of size n = 2 from a simple random sample without replacement. In a SRSWOR, each unit has a probability of n
N- = 2
7 of being in the sample, same probability as SRSWR. Each possible sample in Table ?? is equally likely with probability 1-
42 , because there are 42 possible samples. For each sample of size 2, there are two ways it could be obtained, considering order. Ignoring the order a unit is selected there are (7 2) = 21 possible different samples. Table ?? illustrates SRSWOR with the sampling frame being all units in the target population, or simply the population. Often when sampling, some units in the target population are not in the sample frame and thus have a zero probability of being included in the sample. Again imagine if a six sided die was used to select the units to be sampled. On the second roll of the dice if the number obtained equaled that of the first roll, the die would have to be rolled again, to obtain two distinct units for the sample (SRSWOR). Thus the units labeled 1 to 6 would have a probability of of being in the sample and unit number 7 has a zero probability. Table ?? illustrates the latter situation, with the sampling frame being being units labeled 1 to 6 within the population.

Tables ??, ??, and ?? illustrate unequal probability sampling with and without replacement. Many samples are taken using unequal probability sampling. Again the majority of basic data analysis techniques assumes equal probability sampling was employed. In general, unequal probability sampling with replacement is much easier than without replacement to calculate estimates of the population quantity of interest. The author wishes to warn the reader to consider carefully before deciding on an unequal probability sample without replacement. Keep in mind that software is getting better and better at analyzing complicated sampling designs, in future the latter statement may not be valid. The main reason for the additional complication with unequal probability sampling without replacement is determining the probability of a specific unit will be in the sample. A general formula for calculating the probability unit i is in the sample is:

∑ P (i ∈ s) = P (s) i∈s,s∈S

This formula could be used for SRSRWR or SRSWOR. For example, a SRSWR of size n = 2 from 6 out of the 7 units, Table ??, the probability of selecting unit 5 equals

P(i = 5 s) = ∑ _5s,sP(s)
= P(1, 5) + P(2, 5) + P(3, 5) + P(4, 5) + P(5, 5) + P(6, 5) + P(7, 5)
= + + + + + + 0
=

An example with unequal probability sampling without replacement, Table ??, the probability of selecting unit 5 equals

P(i = 5 s) = ∑ _5s,sP(s)
= P(1, 5) + P(2, 5) + P(3, 5) + P(4, 5) + P(6, 5) + P(7, 5)
= 0.021 + 0.057 + 0.010 + 0.064 + 0.010 + 0.092
= 0.254

Table 5.1:

Imaginary Population of Size N = 7


unit label	1	2	3	4	5	6	7
y-value	56	60	45	65	55	46	72

Table 5.2:

All Possible Samples of Size n = 2 with SRSWR


(1,1)	(1,2)	(1,3)	(1,4)	(1,5)	(1,6)	(1,7)
(2,1)	(2,2)	(2,3)	(2,4)	(2,5)	(2,6)	(2,7)
(3,1)	(3,2)	(3,3)	(3,4)	(3,5)	(3,6)	(3,7)
(4,1)	(4,2)	(4,3)	(4,4)	(4,5)	(4,6)	(4,7)
(5,1)	(5,2)	(5,3)	(5,4)	(5,5)	(5,6)	(5,7)
(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,6)	(6,7)
(7,1)	(7,2)	(7,3)	(7,4)	(7,5)	(7,6)	(7,7)

Table 5.3:

SRSWR and Sampling Frame All Units


Sample	P(s)		P(s)	Sample	P(s)		P(s)

(1,1)	1/49	56.0	1.14	(3,4)	2/49	55.0	2.24
(1,2)	2/49	58.0	2.37	(3,5)	2/49	50.0	2.04
(1,3)	2/49	50.5	2.06	(3,6)	2/49	45.5	1.86
(1,4)	2/49	60.5	2.47	(3,7)	2/49	58.5	2.39
(1,5)	2/49	55.5	2.27	(4,4)	1/49	65.0	1.33
(1,6)	2/49	51.0	2.08	(4,5)	2/49	60.0	2.45
(1,7)	2/49	64.0	2.61	(4,6)	2/49	55.5	2.27
(2,2)	1/49	60.0	1.22	(4,7)	2/49	68.5	2.80
(2,3)	2/49	52.5	2.14	(5,5)	1/49	55.0	1.12
(2,4)	2/49	62.5	2.55	(5,6)	2/49	50.5	2.06
(2,5)	2/49	57.5	2.35	(5,7)	2/49	63.5	2.59
(2,6)	2/49	53.0	2.16	(6,6)	1/49	46.0	0.94
(2,7)	2/49	66.0	2.69	(6,7)	2/49	59.0	2.41
(3,3)	1/49	45.0	0.92	(7,7)	1/49	72.0	1.47

				E[ ]	= 57.00
				Bias( )	= 0.00

Table 5.4:

SRSWR and Sampling Frame Units 1-6


Sample	P(s)		P(s)	Sample	P(s)		P(s)

(1,1)	1/36	56.0	1.56	(3,4)	1/18	55.0	3.06
(1,2)	1/18	58.0	3.22	(3,5)	1/18	50.0	2.78
(1,3)	1/18	50.5	2.81	(3,6)	1/18	45.5	2.53
(1,4)	1/18	60.5	3.36	(3,7)	0	58.5	0.00
(1,5)	1/18	55.5	3.08	(4,4)	1/36	65.0	1.81
(1,6)	1/18	51.0	2.83	(4,5)	1/18	60.0	3.33
(1,7)	0	64.0	0.00	(4,6)	1/18	55.5	3.08
(2,2)	1/36	60.0	1.67	(4,7)	0	68.5	0.00
(2,3)	1/18	52.5	2.92	(5,5)	1/36	55.0	1.53
(2,4)	1/18	62.5	3.47	(5,6)	1/18	50.5	2.81
(2,5)	1/18	57.5	3.19	(5,7)	0	63.5	0.00
(2,6)	1/18	53.0	2.94	(6,6)	1/36	46.0	1.28
(2,7)	0	66.0	0.00	(6,7)	0	59.0	0.00
(3,3)	1/36	45.0	1.25	(7,7)	0	72.0	0.00

				E[ ]	= 54.50
				Bias( )	= -2.50

Table 5.5:

All Possible Samples of Size n = 2 with SRSWOR


(1,2)	(1,3)	(1,4)	(1,5)	(1,6)	(1,7)
(2,1)	(2,3)	(2,4)	(2,5)	(2,6)	(2,7)
(3,1)	(3,2)	(3,4)	(3,5)	(3,6)	(3,7)
(4,1)	(4,2)	(4,3)	(4,5)	(4,6)	(4,7)
(5,1)	(5,2)	(5,3)	(5,4)	(5,6)	(5,7)
(6,1)	(6,2)	(6,3)	(6,4)	(6,5)	(6,7)
(7,1)	(7,2)	(7,3)	(7,4)	(7,5)	(7,6)

Table 5.6:

SRSWOR and Sampling Frame All Units


Sample	P(s)		P(s)

(1,2)	1/21	58.0	2.76
(1,3)	1/21	50.5	2.40
(1,4)	1/21	60.5	2.88
(1,5)	1/21	55.5	2.64
(1,6)	1/21	51.0	2.43
(1,7)	1/21	64.0	3.05
(2,3)	1/21	52.5	2.50
(2,4)	1/21	62.5	2.98
(2,5)	1/21	57.5	2.74
(2,6)	1/21	53.0	2.52
(2,7)	1/21	66.0	3.14
(3,4)	1/21	55.0	2.62
(3,5)	1/21	50.0	2.38
(3,6)	1/21	45.5	2.17
(3,7)	1/21	58.5	2.79
(4,5)	1/21	60.0	2.86
(4,6)	1/21	55.5	2.64
(4,7)	1/21	68.5	3.26
(5,6)	1/21	50.5	2.40
(5,7)	1/21	63.5	3.02
(6,7)	1/21	59.0	2.81

E[ ]	= 57.00
Bias( )	= 0.00

Table 5.7:

SRSWOR and Sampling Frame Units 1 to 6


Sample	P(s)		P(s)

(1,2)	1/15	58.0	3.87
(1,3)	1/15	50.5	3.37
(1,4)	1/15	60.5	4.03
(1,5)	1/15	55.5	3.70
(1,6)	1/15	51.0	3.40
(1,7)	0	64.0	0.00
(2,3)	1/15	52.5	3.50
(2,4)	1/15	62.5	4.17
(2,5)	1/15	57.5	3.83
(2,6)	1/15	53.0	3.53
(2,7)	0	66.0	0.00
(3,4)	1/15	55.0	3.67
(3,5)	1/15	50.0	3.33
(3,6)	1/15	45.5	3.03
(3,7)	0	58.5	0.00
(4,5)	1/15	60.0	4.00
(4,6)	1/15	55.5	3.70
(4,7)	0	68.5	0.00
(5,6)	1/15	50.5	3.37
(5,7)	0	63.5	0.00
(6,7)	0	59.0	0.00

E[ ]	= 54.50
Bias( )	= -2.50

Table 5.8:

Imaginary population of size N = 7 from Table ??, for unequal probability sampling with probability of selection of unit i equaling p_i.


unit label	1	2	3	4	5	6	7
y-value	56	60	45	65	55	46	72
p_i	0.08	0.20	0.04	0.22	0.12	0.04	0.30

Table 5.9:

Unequal Probability Sampling WR and


Sample	P(s)		P(s)	Sample	P(s)		P(s)

(1,1)	0.006	56.0	0.36	(3,4)	0.018	55.0	0.97
(1,2)	0.032	58.0	1.86	(3,5)	0.010	50.0	0.48
(1,3)	0.006	50.5	0.32	(3,6)	0.003	45.5	0.15
(1,4)	0.035	60.5	2.13	(3,7)	0.024	58.5	1.40
(1,5)	0.019	55.5	1.07	(4,4)	0.048	65.0	3.15
(1,6)	0.006	51.0	0.33	(4,5)	0.053	60.0	3.17
(1,7)	0.048	64.0	3.07	(4,6)	0.018	55.5	0.98
(2,2)	0.040	60.0	2.40	(4,7)	0.132	68.5	9.04
(2,3)	0.016	52.5	0.84	(5,5)	0.014	55.0	0.79
(2,4)	0.088	62.5	5.50	(5,6)	0.010	50.5	0.48
(2,5)	0.048	57.5	2.76	(5,7)	0.072	63.5	4.57
(2,6)	0.016	53.0	0.85	(6,6)	0.002	46.0	0.07
(2,7)	0.120	66.0	7.92	(6,7)	0.024	59.0	1.42
(3,3)	0.002	45.0	0.07	(7,7)	0.090	72.0	6.48

				E[ ]	= 62.62
				Bias( )	= 5.62

Table 5.10:

Unequal Probability Sampling WOR and


Sample	P(s)		P(s)

(1,2)	0.037	58.0	2.17
(1,3)	0.007	50.5	0.34
(1,4)	0.042	60.5	2.52
(1,5)	0.021	55.5	1.18
(1,6)	0.007	51.0	0.35
(1,7)	0.060	64.0	3.86
(2,3)	0.018	52.5	0.96
(2,4)	0.111	62.5	6.96
(2,5)	0.057	57.5	3.29
(2,6)	0.018	53.0	0.97
(2,7)	0.161	66.0	10.61
(3,4)	0.020	55.0	1.12
(3,5)	0.010	50.0	0.52
(3,6)	0.003	45.5	0.15
(3,7)	0.030	58.5	1.73
(4,5)	0.064	60.0	3.83
(4,6)	0.020	55.5	1.13
(4,7)	0.179	68.5	12.25
(5,6)	0.010	50.5	0.53
(5,7)	0.092	63.5	5.86
(6,7)	0.030	59.0	1.75

E[ ]	= 62.12
Bias( )	= 5.12

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