Final answer:
Small sample sizes are acceptable in specific scenarios, such as proving the existence of black swans, studying treatments for rare diseases, and discovering new fundamental particles, due to their groundbreaking nature or the limitations in available subjects.
Step-by-step explanation:
The question asks about scenarios where a small sample size is acceptable. It's important to recognize that while larger samples are generally preferable because they tend to be more reliable, there are situations where small sample sizes are inevitable or sufficient for drawing conclusions:
- Discovering that there are indeed black swans (a): A single sighting can confirm the existence of black swans, thus a small sample size here is significant because it disproves the presumption that all swans are white.
- The effect of a treatment on patients with a rare disease (b): Due to the rarity of the disease, the sample pool may be limited, making a small sample size necessary and acceptable.
- Finding a new kind of fundamental particle (d): Particles may be difficult to detect or produce, therefore scientists often work with very small numbers of these particles, and even a single observation can be groundbreaking.
As for the other options, c) polling voters about political beliefs and e) determining the effectiveness of a new drug typically require larger sample sizes to ensure that the results are representatively accurate and statistically significant.