Final answer:
The smallest sample size could be c) 30, which aligns with the common threshold for large sample size in hypothesis testing and enables the use of normal distribution approximation according to the Central Limit Theorem. Option C is correct.
Step-by-step explanation:
The minimum sample size required to meet the conditions for a hypothesis test depends on certain criteria. For normally distributed populations, you can use a t-test when the sample size is small and the population variance is unknown. However, if the sample size is large (typically n ≥ 30 according to the Central Limit Theorem), you can use the normal distribution to perform the hypothesis test. In our scenarios:
The sample size of 20, with a population mean of 13, sample mean of 12.8, and standard deviation of 2, assuming the population is normal, suggests the use of a t-distribution for performing the hypothesis test.
The sample size of 10 in each of two groups for an independent-samples t-test means you must use the t-distribution.
With a sample size of 108 and known population standard deviation, the z-distribution would be used for hypothesis testing.
Given these scenarios and guidelines, the smallest sample size could be c) 30, which is a common threshold for considering the sample size large enough to use normal distribution approximations for hypothesis testing according to the Central Limit Theorem.