Suppose that for a particular family of T-distributions that t2 has 2 degrees of freedom (df) and t9 has 9 degrees of freedom. What would you anticipate happening to tdf as the df increase?

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BlackTom AI · Accepted Answer

Understanding the behavior of the t-distribution as degrees of freedom change is key here.
- Concept: The t-distribution with df degrees of freedom is similar to the standard normal distribution, but has heavier tails. As df increases, the t-distribution becomes more like the standard normal distribution (Z). This is because the additional variability from estimating the standard deviation from the sample decreases with more data, reducing tail heaviness.
- Application to the question: Given t2 (df = 2) and t9 (df = 9), you would expect the distribution to become increasingly closer to the standard normal as df grows beyond 9. In the limit of large df, t_df converges to Z, the standard normal distribution.
- If there were options, a correct choice would state something like: 't_df approaches the standard normal distribution (Z) as df increases.' This reflects the known convergence property of the t-distribution.
- Why other interpretations would be incorrect (in general): If an option claimed that t_df remains significantly different from Z for all df, that would contradict the established convergence. If an option claimed t_df becomes more spread out than Z with larger df, that would misstate the tail behavior, since tails become lighter and approach normality. Since no alternative options are provided here, we focus on the core idea that increasing degrees of freedom makes the t-distribution resemble the standard normal more closely, ultimately approaching Z.

STAT 1450.02 SP2025 (28132) Checkpoint 20.1 & 20.2

Suppose that for a particular family of T-distributions that t2 has 2 degrees of freedom (df) and t9 has 9 degrees of freedom. What would you anticipate happening to tdf as the df increase?

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