Final answer:
Caution in interpreting t-test results is warranted when the population distribution significantly deviates from normality and the sample size is less than 20, as small samples require a closer adherence to the normality assumption. A large sample size of more than 30 allows for the central limit theorem to ensure the approximation to normality. For sample sizes of less than 10, caution is also advised due to inadequate data.
Step-by-step explanation:
In statistical analysis, caution should be exercised when interpreting results from a t-test if the population distribution departs substantially from normality and the sample size is less than 20. This scenario is of particular concern because the t-test assumes that the population from which the sample is drawn is normally distributed, especially when dealing with small sample sizes. The more the sample size decreases, the more critical the assumption of normality becomes. If the sample is small and the population distribution is not normal, the t-test may not be reliable.
A t-test can be used confidently when the population distribution is normal or when the sample size is large, typically greater than 30. When the sample size is large, the central limit theorem applies, suggesting that the sampling distribution of the sample mean will be approximately normal even if the population distribution is not. Therefore, the distribution needed for a hypothesis test, like a t-test, is the Student's t-distribution when the population standard deviation is unknown and the sample standard deviation is used as an estimate.
However, if the sample size is less than 10, caution must also be taken, as such a small sample size may not provide enough data to reliably approximate the population distribution, whether normal or not.