Degrees of freedom influence the probability distributions of which statistics mentioned in the material?

• A. F-statistic
• B. t-statistic
• C. Chi-square statistic
• D. All of the above

Answer: (D)

Degrees of freedom determine the exact sampling distribution used for a statistic, and that shape governs how likely different outcomes are under the null hypothesis. For the t-statistic, the distribution is a t distribution whose shape depends on its degrees of freedom. With few degrees of freedom, the t distribution is wider and has heavier tails, which affects p-values and critical values; as df grows, it approaches a standard normal distribution.

For the F-statistic, the distribution is the F distribution and is defined by two degrees of freedom: one for the numerator and one for the denominator. These two dfs determine the exact shape and tail behavior, so the critical values and p-values depend on the specific df pair.

For the chi-square statistic, the distribution is the chi-square distribution with a degrees-of-freedom parameter equal to the number of independent pieces of information being summed. Lower degrees of freedom make the distribution highly skewed; higher degrees of freedom make it more symmetric and closer to normal. This df sensitivity changes how extreme observed values are judged.

Because degrees of freedom influence the probability distributions of all three statistics, the correct answer is that all of the above.