Final answer:
The t-distribution's shape is indeed a function of sample size, as it changes with the degrees of freedom, which increase with larger sample sizes.
Step-by-step explanation:
The answer to the question about the characteristics of the t-distribution is that (a) The shape of the distribution is a function of sample size is true. The t-distribution varies depending on the degrees of freedom, which are directly related to the sample size. As the sample size increases, the degrees of freedom increase, and the t-distribution becomes closer to a normal distribution. Therefore, the distribution of Z, which is the standard normal distribution, is only identical to the t-distribution when the degrees of freedom are large. The t-distribution is not bi-modal; it is unimodal with a single peak, just like the normal distribution, though it has heavier tails when the sample size is small. As the sample size increases, these tails become lighter, and the distribution looks more like the bell-shaped normal curve. It's also important to note that the t-distribution curve always approaches but never touches the x-axis.