Final answer:
The false statement about degrees of freedom is that the graph of the Student's t distribution looks more like the standard normal distribution as df decrease; in reality, it more closely resembles the normal distribution as df increase.
Step-by-step explanation:
The statement that is FALSE is: "As the degrees of freedom decrease, the graph of the Student's t distribution looks more like the graph of the standard normal distribution." In fact, the opposite is true. As the number of degrees of freedom (df) increases, the t-distribution approaches the shape of the standard normal distribution. Here are clarifications for each statement:
- There is indeed a different t-distribution for every different degrees of freedom, which is true.
- The shape of the sampling distribution of t depends on the degree of freedom, which is also true.
- The actual amount of variability in the sampling distribution of t depends on the degree of freedom, which is true.
- The degree of freedom is a measure of the number of independent pieces of information that can be used to estimate a population parameter, which is true.
To clarify, the statement about the t-distribution approaching the normal distribution as degrees of freedom decrease is incorrect. The degrees of freedom (df), typically represented as n - 1, are the sample size minus 1. The properties of the Student's t-distribution indicate that as df increases, the graph initiates to look more symmetrical and similar to the standard normal curve, with less variability in its tails.