| 5.7 | Sampling With or Without Replacement |
- Key Idea 1: Sampling without replacement means that you do not replace the item selected in the population after each draw...
- Suppose you want to send a signal with four different colored signal flags. You pick one flag and transmit, then you have a choice. You can put that flag back in before you select the second flag, or you can set aside and make your second choice from among the three remaining colors.
- Let’s count the number of signals possible if you sample without replacement (where R = red, O = orange, Y = yellow, and B = blue). See Figure of Sampling without Replacement.
- What is the probability of the signal ORY (O followed by R followed by Y)? Since this signal can occur in only one way, and all sequences are equally-likely, we have p(ORY) = 1/24.
Discussion
- Key Idea 2: Sampling with replacement means that you replace the item selected in the population after each draw...
- If you signal with replacement, the probabilities in the above signal flag example will change.
- There will be 4x4x4 = 64 possible signals, rather than 4x3x2. ORY can still occur in only one way, and hence p(ORY) = 1/64.
- Key Idea 3: If the sample is very small relative to the population, sampling with replacement yields a result virtually equivalent to sampling without replacement ...
- In practical situations, sampling is typically done without replacement.
- Making inferences from a sample usually requires statistical procedures that assume sampling with replacement, but in cases where the sample is very small relative to the population this is okay. See Figure of Sampling with Replacement.
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