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**Unformatted text preview: **Statistics 651: Survey Sampling Methods
Winter 2011
Practice Problems for Exam 2 Solutions
1. Nursery Workers
(a) This is stratied random sampling followed by a single stage cluster sample.
The probability that any one worker is selected can be calculated:
n
πij = P(worker selected| nursery selected)P(nursery selected) = 1 × Nh
h
Thus, for each of the strata, the weight of every worker in that stratum is
as shown in this table:
State Nh nh weight = Nh /nn
AZ
243 20
12.15000
TN 1383 23
60.13043
RI
87
2
43.50000
DE
131 10
13.10000
OH 1434 32
44.81250
MI 1525 12
127.08333
IN
437 10
43.70000
FL 7703 30
256.76667
Nh
,
nh
(b) With the further subsampling, the probability of selection now becomes:
n
m
πij = P(worker selected| nursery selected)P(nursery selected) = Mih × Nh
ih
h
Thus, the weight is wij =
For the rst nursery:
Nh Mih
.
nh mih
wij =
87 × 21
= 91.35
2 × 10
wij =
87 × 34
= 147.9
2 × 10
For the second nursery:
Note that the weight increased over 43.5 for both nurseries, and it increased
more for the larger nursery. This is because the 10 people selected from the
larger nursery need to represent more people than the 10 people from the
smaller nursery.
2. Lockhart City
(a) I used the SAS program using sampling probability proportional to size. First,
I read the data from gure A2 into SAS, and then I used SURVEYSELECT
with PPS, and SIZE = number of households in the county.
I have chosen districts 51, 71, 62, 65, and 63.
1
(b) P(select household ij on rst draw) =
P(select household j on rst draw | Cluster i selected on rst draw) ×
P(Cluster i selected on rst draw) =
1 Mi
1
1
= K = 19664 .
Mi K
(c) Yes.
(d) Variance Estimates and CIs (calculated in R):
i. de = 3. Ignoring the fpc for the SRS sample,
ˆ
V (SRS ) = 0.19
ˆ
ˆ
V (Complex) = def f V (SRS ) = 3 0.19 = 0.575
95% CI:
11.047 ± 1.96 0.575 = (9.55, 12.54)
2
...

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