Q 22. - Elementary Statistics

Question

Water Quality. In the article "Randomized Stratified Sampling Methodology for Water Quality in Distribution Systems" (Journal of Water Resources Planning and Management, Vol. 130, Issue 4, pp. 330-338), V. Speight et al. proposed the method of stratified sampling to collect water samples for water-quality testing.

The following table separates the Durham, North Carolina water distribution system into strata based on distance from the nearest treatment plant.

Use the table to design a procedure for obtaining a stratified sample (with proportional allocation) of 80 water samples from Durham.

Hint: Refer to the remarks about strata sample size on pages 22-23 following Example 1.14.

Step-by-Step Solution

Answer

Region	Sample	Size Sample
Less than 1.5 miles	9	378, 585, 590, 411, 981, 407, 639, 49, 757
1.5 - Less than 3.0 miles	22	2173, 1524, 2555, 4411, 2290, 2889, 3204, 3258, 1767, 2892, 1724, 2769, 4268, 2114, 2392, 2042, 1411, 4160, 2311, 3076,  2412, 2254
3.0 - Less than 4.5 miles	19	5630, 5235, 4759, 5632, 6646, 5700,  5415, 5224, 6585, 5728, 6285, 6566, 6788, 4860, 5394, 6323, 4941, 5921,  5030
4.5 - Less than 6.0 miles	11	5630   5235   4759   5632   6646   5700   5415   5224   6585   5728   6285   6566   6788   4860   5394   6323   4941   5921   5030
6.0 - Less than 7.5 miles	9	7368, 8702, 8641, 7909, 8632, 7690, 8459   8546, 7856, 8234, 8424
7.5 miles or greater	10	11684, 11166, 11692, 11523, 11425, 11235,  11005, 10556, 11643, 11124

Step 1. Given information.

The given statement is:

The following table separates the Durham, North Carolina water distribution system into strata based on distance from the nearest treatment plant.

Step 2. Calculate the sample from each stratum.

The sample size for the stratum "Less than 1.5 miles":

$$\begin{gathered} \frac{1,310}{11,707}\times 80=8.95 \\ \approx 9 \end{gathered}$$

The sample size for the stratum "1.5 - Less than 3.0 miles":

$$\begin{gathered} \frac{3,166}{11,707}\times 80=21.63 \\ \approx 22 \end{gathered}$$

The sample size for the stratum "3.0 - Less than 4.5 miles":

$$\begin{gathered} \frac{2,825}{11,707}\times 80=19.3 \\ \approx 19 \end{gathered}$$

The sample size for the stratum "4.5 - Less than 6.0 miles":

$$\begin{gathered} \frac{1,593}{11,707}\times 80=10.88 \\ \approx 11 \end{gathered}$$

The sample size for the stratum "6.0 - Less than 7.5 miles":

$$\begin{gathered} \frac{1,350}{11,707}\times 80=9.22 \\ \approx 9 \end{gathered}$$

The sample size for the stratum " 7.5 miles or greater":

$$\begin{gathered} \frac{1,463}{11,707}\times 80=9.99 \\ \approx 10 \end{gathered}$$

Step 3. Use a random number generator to obtain a sample from each stratum.

We have used a random sample generator named MINITAB to get a simple random sample of size 9 from the 1310 numbers of stratum 1 "Less than 1.5 miles" and obtained numbers:

378   585   590   411   981   407   639   49   757

Similarly, we have found the sample size for other strata:

Stratum 2 ("1.5 - Less than 3.0 miles"):

2173   1524   2555   4411    2290   2889    3204    3258    1767    2892    1724    2769     4268    2114    2392    2042    1411   4160    2311    3076    2412   2254

Stratum 3 ("3.0 - Less than 4.5 miles"):

5630   5235   4759   5632   6646   5700   5415   5224   6585   5728   6285   6566   6788   4860   5394   6323   4941   5921   5030

Stratum 4 ("4.5 - Less than 6.0 miles"):

7368   8702   8641   7909   8632   7690   8459   8546   7856   8234   8424

stratum 5 ("6.0 - Less than 7.5 miles"):

9434    9923    9460    9654    9540    9176    9273    9721    9382

Stratum 6 ("7.5 miles or greater"):

11684    11166    11692    11523    11425    11235    11005     10556    11643    11124