To solve these problems, we need to refer to a t-distribution table or use a calculator that provides critical values for a t-distribution.

Problem 1:

Given:

• Degrees of freedom ($df$) = 18
• Significance level ($\alpha$) = 0.01

To find $t_{0.01}(18)$, we are looking for the t-value that has a right-tail probability of 0.01 with 18 degrees of freedom.

Using a t-distribution table or calculator, we find: $t_{0.01}(18) \approx 2.552$

Problem 2:

Given:

• Degrees of freedom ($df$) = 19
• Right-tail probability = 0.94 (or left-tail probability = 0.06)

To find $t_{0.94}(19)$, we need the t-value with a left-tail probability of 0.06 (since 1 - 0.94 = 0.06).

Using a t-distribution table or calculator, we find: $t_{0.94}(19) \approx -1.440$

Final Answers:

1. $t_{0.01}(18) = 2.552$
2. $t_{0.94}(19) = -1.440$

Would you like further details on how these values were found, or any additional help?

Here are some follow-up questions related to this topic:

1. How would the critical t-value change with a larger sample size?
2. What is the relationship between the t-distribution and the normal distribution?
3. How can you use these critical values in hypothesis testing?
4. What happens to the t-distribution shape as degrees of freedom increase?
5. How do you find critical values for two-tailed tests?

Tip: In hypothesis testing, choosing the right significance level is key to balancing Type I and Type II errors.