Question

Mickey wants to survey the members of her gym to gauge what classes she should offer next month.
There are 450 total gym members who attend classes. What is the least number of people that Mickey
should include in her survey?

Answer 1

I don’t have enough information to give a single correct number — what do you want the survey to guarantee?

Two common interpretations and their answers:

1) Pigeonhole-style guarantee (combinatorics)
- If you want to guarantee at least k people with the same property among m possible categories, you need (k−1)m + 1 people. I can apply this if you tell me how many categories and what k you want guaranteed.

2) Statistical-sample-size (estimate a proportion to a given confidence and margin of error)
- Use the usual formula with finite-population correction:
n = (N z^2 p(1−p)) / ((N−1) ME^2 + z^2 p(1−p))
where N=450, z depends on confidence (e.g. 1.96 for 95%), p is estimated proportion (use 0.5 for worst case), and ME is desired margin of error.
- Example: for 95% confidence and ±5% margin of error, p=0.5 gives n ≈ 208.

Tell me which interpretation (or the missing details: number of categories and k, or confidence level/margin of error/estimated p) and I’ll compute the exact least number.