Final answer:
The distributions for known sigma tend to be normal distributions, while those for unknown sigma, with small sample sizes, are t-distributions with thicker tails. As sample sizes increase, the t-distribution approaches the normal distribution. Skewness or peakness of a distribution does not solely depend on sigma's status.
Step-by-step explanation:
When comparing distributions with unknown sigma (σ) to those with known sigma, the distribution characteristics can change based on sample size and whether we are estimating sigma using sample data. Specifically, the correct answer to the question is:
C) More peaked and narrower due to larger sample sizes.
This choice does not actually reflect how distributions behave with unknown vs. known sigma. When sigma is unknown and the sample size (n) is small, the distribution of the sample mean is best described using a t-distribution, which tends to have more probability in its tails and is less condensed near the mean compared to the normal distribution, which is used when sigma is known. As the sample size increases, the t-distribution approaches the normal distribution, meaning it becomes less peaked and narrower (option C is the opposite). However, neither the skewness nor the peakness of the distribution is simply determined by unknown or known sigma solely; they depend on multiple factors including sample size and the underlying population distribution.
Statements regarding skewness and the chi-square distribution relate to characteristics of the respective distributions and do not directly answer the question posed about sigma being unknown versus known. Similarly, facts about the F distribution describe its skewness and use in ANOVA tests, which are outside the scope of this question. Instead, a key concept is that with large enough sample sizes, the Central Limit Theorem asserts that the sampling distribution will approach normality regardless of sigma's status.