Final answer:
For small sample sizes, the appropriate distribution to use is the Student's t-distribution, not binomial or normal. The Poisson distribution can approximate a binomial distribution in the case of large n and small p, and the exponential distribution fits the provided example of running distances.
Step-by-step explanation:
Understanding Distributions for Small Sample Sizes
When dealing with small sample sizes, it's important to choose the correct distribution to analyze data. If a sample size is too small, the normal distribution is not always the best choice due to the Central Limit Theorem's requirement of a sufficiently large sample size. Instead, the Student's t-distribution should be used, especially when sample sizes are too small (typically n < 30) and population standard deviation is unknown. The t-distribution takes into account the additional variability due to the small sample size, providing a more accurate assessment of uncertainty.
In the case of a binomial distribution scenario, if the number of trials n is large and the probability of success p is small, a Poisson distribution can sometimes be used as an approximation. Therefore, for a hypothesis test or when calculating probabilities for rare events in a large number of trials, the Poisson distribution may be more appropriate than the normal distribution.
As for the example given, describing that fewer people can run as distance increases, it best follows an exponential distribution, which models the time or distance until an event of interest occurs. The correct option for this example is therefore C. exponential.
Please note: For binomial distributions, if np and nq are both greater than five, the distribution can be approximated by a normal distribution, but this is usually for larger sample sizes, not small sample sizes as the central question implies.
In summary, for very small sample sizes, it is generally best to use the Student's t-distribution. For large numbers of trials with a small probability of success, the Poisson distribution is often used to approximate a binomial distribution, and for the given example, the exponential distribution is the correct choice.