Environmental Engineering Reference
In-Depth Information
number generator at http://www.random.org. The random numbers shown below are obtained
from the Tool | Data Analysis | Random Number Generation function using Excel. Readers
should be aware of the fact that your answers will be different because of the randomization
process each time you use it.
For the large tank, the two-dimensional coordinates can be denoted by a radius (25 ft)
and a circumference (2p25 ¼ 157 ft). Hence, 15 pairs of random numbers are
generated for the circumference coordinate from 0 to 157, and radius coordinate in the
range of 0-25:
(1, 1), (25, 5), (130, 21), (15, 3), (156, 25), (26, 5), (107, 17), (102, 16), (138, 22), (20, 4),
(18, 4), (138, 22), (27, 5), (92, 15), (124, 20).
For the overflow small tank, 15 random numbers between 0 and 300 min are needed, and they
are: 2, 155, 31, 126, 127, 168, 62, 28, 108, 18, 41, 261, 238, 18, 120.
EXAMPLE 3.3.
Calculating means and standard deviation for a stratified random
sampling. Studies were conducted to test lead (Pb) concentrations in surface soils due to
atmospheric deposition of emissions from a smelter. Prior information revealed that Pb is
higher in the prevailing downwind direction (from East to West). The Pb concentration is also
higher in clayed soils than sandy soils. It was determined that a stratified random sampling
approach was appropriate. A total of 30 samples were collected, and the number of samples in
each stratum was proportionally allocated based on the estimated percentage land area under
the specified wind direction. The stratum weights and the resulting statistics of analytical
results are shown in Table 3.2, and the raw data are shown in the footnotes. Estimate the
overall means and standard deviation.
SOLUTION:
Use Eqs 3.2 and 3.3 to calculate the overall mean and variance as follows.
x ¼
X
r
w
k
x
k
¼ 0:579:0þ0:265:5þ0:1756:8þ0:1350:0 ¼ 68:8mg=kg
k¼1
¼
X
r
w
k
s
k
s
2
n
k
¼ð0:5
2
7:0
2
=15þ0:2
2
3:4
2
=6þ0:17
2
2:9
2
=5þ0:13
2
2:6
2
=4Þ
k¼1
2
¼ 0:972ðmg=kgÞ
1=2
1=2
s ¼ðs
2
Þ
¼ð0:972Þ
¼ 0:99mg=kg
Table 3.2 Summary statistics for lead contents in four strata of soils in a nearby smelter
a
Sample
Standard
Stratum
Stratum
number (n
k
) Mean(x
k
)
deviation (s
k
) weight (w
k
)
Downwind clayed soil (S1)
15
79.0
7.0
0.5
Downwind sandy soil (S2)
6
65.5
3.4
0.2
Perpendicular wind clayed soil (S3)
5
56.8
2.9
0.17
Perpendicular wind sandy soil (S4)
4
5
0.0
2.6
0.13
Overall
n ¼ 30
x ¼ ?
s ¼ ?
W
k
¼ 1:00
a
Raw data (unit in mg/kg): S1¼80, 75, 89, 65, 73, 77, 74, 83, 82, 85, 76, 87, 77, 90, 72; S2¼66,
68, 65, 60, 70, 64; S3¼60, 55, 59, 57, 53; S4¼53, 51, 49, 47.
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