Page 209 - Statistics for Environmental Engineers

l1592_frame_Ch23 Page 209 Tuesday, December 18, 2001 2:44 PM

TABLE 23.4
Data for the Stratified Sample for Examples 23.9, 23.10, and 23.11
Observations Mean Variance Size of Weight
2
n i y i s i Stratum w i

Stratum 1 20 34 35.4 1500 0.5
Stratum 2 8 25 180 750 0.25
Stratum 3 12 19 12 750 0.25

where n s is the number of strata and the w i are weights that indicate the proportion of the population
included in stratum i. The estimated variance of is:
y

2 2 s n 2
s = w 1 ---- + w 1 ---- + … + w n ------ s
2
2 s 1
2 s 1
2
y s
n 1 n 1 n n
s
Example 23.9
Suppose we have the data in Table 23.4 from sampling a contaminated site that was known to
have three distinct areas. There were a total of 3000 parcels (acres, cubic meters, barrels, etc.)
that could have been sampled. A total of n = 40 observations were collected from randomly
selected parcels within each stratum. The allocation was 20 observations in stratum 1, 8 in stratum
2, and 12 in stratum 3. Notice that one-half of the 40 observations were in stratum 1, which is
also one-half of the population of 3000 sampling units, but the observations in strata 2 and 3 are
not proportional to their populations. This allocation might have been made because of the relative
cost of collecting the data, or because of some expected characteristic of the site that we do not
know about. Or, it might just be an inefficient design. We will check that later.
The overall mean is estimated as a weighted average:

(
(
(
y = 0.5 34) + 0.25 25) + 0.25 19) = 28
The estimated variance of the overall average is the sum of the variances of the three strata
weighted with respect to their populations:


2 180



2 12


2 35.4
s = 0.5 ---------- + 0.25 --------- + 0.25 ------ = 1.9
2
y  20   8   12
The confidence interval of the mean is y ± 1.96 s 2 , or 28 ± 2.7.
y
The confidence intervals for the randomly sampled individual strata are interpreted using
familiar equations. The 95% confidence interval for stratum 2 is y 2 ± 1.96 s 2 /n 2 and 25 ± 1.96
2
180/8 = 25 ± 9.3. This confidence interval is large because the variance is large and the sample
size is small. If this had been known, or suspected, before the sampling was done, a better
allocation of the n = 40 samples could have been made.
Samples should be allocated to strata according to the size of the strata, its variance, and the cost of
sampling. The cost of the sampling plan is:
Cost = Fixed cost + ( c 1 n 1 + c 2 n 2 + … + c n s n n s )
© 2002 By CRC Press LLC