Final answer:
The characteristic that "It approaches z as degrees of freedom decrease" is incorrect for the t distribution. In fact, the t-distribution approaches the z-distribution as the number of degrees of freedom increases and is especially used when sample sizes are small.
Step-by-step explanation:
The statement "It approaches z as degrees of freedom decrease" is not a characteristic of the t distribution. Rather, it is the other way around: the t-distribution approaches the z-distribution (or standard normal distribution) as the number of degrees of freedom increases. The t-distribution is used when the sample size is small, which means when the number of degrees of freedom is lower, the t-distribution has heavier tails compared to the z-distribution. As sample size (n) gets larger, meaning as the degrees of freedom (df) increase, the shape of the t-distribution gets closer to that of the z-distribution. This convergence occurs because the standard error decreases as sample size increases, making the sampling distribution of the sample means closer to a normal distribution.
Additionally, the exact shape of the Student's t-distribution is dependent on the degrees of freedom, and the graph of the Student's t-distribution with a higher degree of freedom will appear more similar to the standard normal distribution curve. To be precise, for each sample size n, there is a specific t-distribution with n - 1 degrees of freedom. The t-distribution is characterized by having more probability in its tails and less near the mean compared to the standard normal distribution. It is important to note that the t-distribution is symptomatic about zero with a mean of zero.