The choice between using a t-distribution table or a z-distribution table depends on a few key factors, particularly:

1. Sample size:

   • Use the t-distribution when the sample size is small (typically $n < 30$) and the population standard deviation ($\sigma$) is unknown.
   • Use the z-distribution when the sample size is large (typically $n \geq 30$) or when the population standard deviation is known.

2. Population standard deviation ($\sigma$):

   • t-distribution is used when the population standard deviation is unknown and must be estimated from the sample data.
   • z-distribution is used when the population standard deviation is known.

3. Shape of the distribution:

   • The t-distribution is more spread out (with fatter tails) than the z-distribution, which accounts for the added uncertainty of estimating the population standard deviation from the sample.
   • As the sample size increases, the t-distribution approaches the z-distribution (standard normal distribution), meaning they become nearly indistinguishable when $n$ is large.

Summary:

• t-distribution: Small sample size, unknown population standard deviation.
• z-distribution: Large sample size, known population standard deviation.

Would you like more details or have any questions on how to use these tables?

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Here are 5 related questions:

1. How does the critical value from the t-distribution differ from the z-distribution for a small sample size?
2. What assumptions must be met to use the z-distribution confidently?
3. How does the degree of freedom affect the t-distribution?
4. When does the t-distribution become approximately the same as the z-distribution?
5. What happens if we use a z-distribution table for a small sample size with an unknown population standard deviation?

Tip: The degrees of freedom for a t-distribution are calculated as $n - 1$, where $n$ is the sample size. This affects how "fat" the tails of the distribution are.