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Random Sampling Help (page 2)

By — McGraw-Hill Professional
Updated on Sep 13, 2011

Replacement or Not?

Just after an element of a set is sampled, it can be left in the set, or else it can be removed. If the element is left in the set so it can be sampled again, the process is called sampling with replacement. If the element is removed so it can't be sampled again, the process is called sampling without replacement.

For a finite sample set, if the experiment is continued long enough the set will eventually be exhausted. Table 5-1A shows how this works if the initial sample contains 10 elements, in this case the first 10 letters of the English alphabet. This is presumably what would happen, on a much larger scale, in the hypothetical telephone-number experiment described above. The experimenters would not likely want to count any particular telephone number twice, because that would bias the experiment in favor of the repeated numbers.

If the sample set is infinite – the set of all points on a length of utility cable, for example, or the set of all geographical locations on the surface of the earth – the size of the sample set does not decrease, even if replacement is not done. An infinite set is inexhaustible if its elements are picked out only one at a time.

If the elements of a sample set are replaced after sampling, the size of the sample set remains constant, whether it is finite or infinite. Table 5-1B shows how this works with the first 10 letters of the English alphabet. Note that once an element has been replaced in a finite sample set, that element may (and almost certainly will) be sampled more than once during the course of the experiment. This happens in the scenario shown by Table 5-1B.

Minimizing Error

When conducting a real-world statistical experiment, errors are inevitable. But there are ways in which error can be kept to a minimum. It's important that all experiments be well conceived and well executed. There are various ways in which an experiment can be flawed. The most common sources of this type of error, which we might call experimental defect error, include:

  • a sample that is not large enough
  • a sample that is biased
  • replacing elements when they should not be replaced
  • failing to replace elements when they should be replaced
  • failing to notice and compensate for factors that can bias the results
  • attempting to compensate for factors that don't have any real effect
  • sloppy measurement of quantities in the sampling process

Improper tallying of the results of a poll or election is a good example of sloppy measurement. If people do not observe the results correctly even though the machines are working the way they ought, it is human error. An election is an example of a digital process. A voter casts either a "yes" (logic 1) or "no" (logic 0) for each candidate.

Measurement error can occur because of limitations in analog hardware. Suppose we want to determine the average current consumed by commercially manufactured 550-watt sodium-vapor lamps when they are operated with only 90 volts rather than the usual 120 volts. We need an alternating-current (AC) ammeter (current-measuring meter) in order to conduct such an experiment. If the ammeter is defective, the results will be inaccurate. No ammeter is perfectly accurate, and with analog ammeters, there is the additional human-error problem of visual interpolation. Figure 5-3 shows what an analog AC ammeter might read if placed in the line in series with a high-wattage, 120-volt lamp operating at only 90 volts.

Minimizing Error

Fig. 5-3. Visual interpolation of an analog meter reading is always subject to error. Illustration for Practice 2 through 4.

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