Choose a random number uniformly from 1 to 10 and survey every 10th student on the list starting from the random number. True or False: each individual has the same chance of being chosen.

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BlackTom AI · Accepted Answer

This question is about a systematic sampling scheme with a random start.

First, imagine the population arranged in a list and we pick a starting position uniformly at random from positions 1 through 10. After choosing the start, we select every 10th person: start, start+10, start+20, and so on.

Consider any particular individual in the list. They will be selected if and only if their position i satisfies i ≡ start (mod 10). Since start is equally likely to be any of the 10 values 1 through 10, exactly one of these congruences will match i for a given i. Therefore, for a sufficiently long list (or in aggregate across many lists), each person has a 1/10 chance of being included in the sample.

However, there is a caveat at the tail end of a finite list. If the list ends before we reach some of the planned sampling points (e.g., i + k*10 exceeds the list length), those positions are not part of the population, so the practical probability of being sampled can become slightly imbalanced for individuals near the end of the list depending on the total list length. In the common theoretical framing where the list is long enough and the sampling continues until the end, the residues mod 10 are represented uniformly, giving equal chances to all individuals.

Summarizing the intuition: with a random start among 10 possible positions and then taking every 10th item, each member of the population has the same probability of being selected in the idealized, sufficiently long list scenario. In real finite lists, there can be minor edge effects near the end, but the intended principle is that the sampling scheme distributes equal probability across individuals.

CIS-124-HYA-CRN48380 Section 15a Practice

Choose a random number uniformly from 1 to 10 and survey every 10th student on the list starting from the random number. True or False: each individual has the same chance of being chosen.

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