Final answer:
The correct answers are b,c and e. The t-distribution is bell-shaped and has a mean of zero. It is not the standard normal distribution and does not have a standard deviation of one, but rather it increases with the degrees of freedom. There are an infinite number of t-distributions, varying by the degrees of freedom.
Step-by-step explanation:
When considering the properties of the t-distribution, several characteristics stand out:
- The t-distribution is bell-shaped, similar to the normal distribution, making this statement true.
- Like the standard normal distribution, the mean of the t-distribution is zero, so this statement is also true.
- The t-distribution is characterized by its degrees of freedom (df), which depend on the sample size, n - 1. As df increases, the t-distribution approaches the standard normal distribution, but they are not one and the same; hence, the t-distribution is not known as the standard normal distribution.
- Unlike the standard normal distribution which has a standard deviation of one, the t-distribution typically has a standard deviation that is greater than one due to more probability in its tails, making the statement about having a standard deviation of one false for the t-distribution.
- Finally, as there is not just one t-distribution but rather an infinite number based on the degrees of freedom, it is true that there are infinite t-distributions.
Given these points, the correct options from the provided list are that the t-distribution:
- is a bell-shaped curve (c)
- has an infinite number of distributions (e)