Final answer:
As the size of the sample increases, the shape of the sampling distribution of means approaches a normal distribution, which is in line with the Central Limit Theorem. The correct answer is B.
Step-by-step explanation:
When investigating how the size of a sample affects the shape of the sampling distribution of means, we refer to the Central Limit Theorem, a fundamental concept in statistics. The theorem states that as the size of the sample (n) increases, the distribution of the sample means will approach a normal distribution, irrespective of the population's distribution shape, provided that the sample size is sufficiently large.
This phenomenon occurs because the variability of the sample means decreases as the sample size increases, which is indicated by a decrease in the standard deviation of the sampling distribution, also known as the standard error of the mean. This results in the distribution of the sample means becoming more concentrated around the true population mean, taking on the bell-shaped curve characteristic of the normal distribution.
Therefore, the correct answer to the question is b) Approaches a normal distribution. As the number of degrees of freedom increases, which happens when sample size increases, the t-distribution also begins to look more like the standard normal distribution. However, it is incorrect to suggest that the normal distribution should never be used even when we are dealing with smaller samples such as n=15; the usage of the normal distribution depends on the underlying distribution of the data and whether other central limit theorem conditions are met.