The minimum sample size to represent a survey of 1936 students should be:

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BlackTom AI · Accepted Answer

To determine the minimum sample size for representing a survey of 1936 students, we need to consider how sample size relates to desired precision (margin of error) and confidence. Here are the options analyzed one by one:

Option 1: 242 students
- This size is within a reasonable range for a finite population of nearly 2000. If we use a common finite-population adjustment formula n = N / (1 + N e^2) with N = 1936, the implied margin of error e would be approximately sqrt((N / n) − 1) / sqrt(N). Plugging in n = 242 gives e ≈ sqrt((1936/242) − 1) / sqrt(1936) ≈ sqrt(8.0 − 1) / 44 ≈ sqrt(7) / 44 ≈ 2.65 / 44 ≈ 0.060, i.e., about a 6% margin of error. This aligns with a scenario where a margin of error around 6% is acceptable at a standard confidence level, making 242 a plausible minimum sample size under those assumptions.

Option 2: 484 students
- This is roughly double 242. Using the same relation, a sample size of 484 would correspond to a smaller margin of error than about 6% (roughly around 4% to 5%, depending on the exact calculation). If the study requires tighter precision than about 6%, 484 could be chosen. However, it is not the minimum; it exceeds what would be needed for that level of precision.

Option 3: 25 students
- A sample of only 25 from 1936 would yield a very large margin of error, likely well over 15% or more, and would provide unreliable estimates for the entire population. This size is far too small to represent the population adequately, regardless of the confidence level assumed.

Option 4: 45 students
- Similar to 25, a sample of 45 would produce a substantial margin of error, far from acceptable for representative estimates of a population near 2,000. This is not sufficient for generalizing to all 1936 students with standard levels of confidence.

In summary, each option represents a different trade-off between precision (margin of error) and effort (sample size). The smaller options (25 and 45) imply large margins of error and poor representativeness, while 484 implies higher precision than necessary for many surveys. The 242-student choice corresponds to a plausible, smaller-margin assumption that still yields a representative sample without oversampling, making it a common target when a moderate level of precision is acceptable for a population of this size.

The minimum sample size to represent a survey of 1936 students should be:

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