Final answer:
The claim that the t-distribution approaches the standard normal distribution as the sample size decreases is incorrect. It is the increase in the sample size or degrees of freedom that makes the t-distribution resemble the standard normal distribution more closely.
Step-by-step explanation:
The statement "As sample size decreases, the t-distribution approaches the standard normal distribution" is false. It is the other way around; the t-distribution approaches the standard normal distribution as the sample size, or equivalently, the degrees of freedom, increases. The central limit theorem supports this concept by stating that as the sample size grows larger, the sampling distribution of the mean will become approximately normal, which can be represented by the standard normal distribution. Additionally, the shape of the Student's t-distribution is dependent on the number of degrees of freedom: the more degrees of freedom there are (which increase with the sample size), the more the t-distribution resembles the standard normal distribution.
Historically, statisticians would use the normal distribution as an approximation for larger sample sizes but would switch to the Student's t-distribution for sample sizes of 30 or less. Nowadays, with the advent of new technology, the t-distribution is advised to be used in cases where the population standard deviation is unknown and the sample standard deviation is used as an estimate.