To solve this problem, we need to use the properties of the t-distribution for both parts (a) and (b). Here's how to approach each part:

(a) Consider a t-distribution with 21 degrees of freedom. Compute $P(t \geq -1.52)$.

We are asked to find the probability $P(t \geq -1.52)$ for a t-distribution with 21 degrees of freedom. To do this:

1. Use the cumulative distribution function (CDF) for the t-distribution:
   • For $P(t \geq -1.52)$, it is the complement of the CDF at $t = -1.52$, i.e., $P(t \geq -1.52) = 1 - P(t \leq -1.52)$
2. Use a calculator or statistical software to find the CDF value for $t = -1.52$ with 21 degrees of freedom.

Using a t-distribution table or calculator: $P(t \leq -1.52) \approx 0.072$ Thus, $P(t \geq -1.52) = 1 - 0.072 = 0.928$

So, the answer to part (a) is: $P(t \geq -1.52) \approx 0.928$

(b) Consider a t-distribution with 8 degrees of freedom. Find the value of $c$ such that $P(-c < t < c) = 0.99$.

This part asks for the critical value $c$ where 99% of the t-distribution with 8 degrees of freedom lies between $-c$ and $c$.

1. Determine the confidence level: The given probability is 0.99, so this is a two-tailed probability, and we need to find the value of $c$ such that: $P(-c < t < c) = 0.99$ This leaves 0.01 split equally in the tails, so each tail has a probability of 0.005.

2. Find the critical value for the t-distribution: Using a t-table or a calculator for 8 degrees of freedom and a cumulative probability of $0.995$ (since 0.99 + 0.005 = 0.995 for the upper bound), $c \approx 3.355$

Thus, the answer to part (b) is: $c \approx 3.355$

Summary:

• (a) $P(t \geq -1.52) \approx 0.928$
• (b) $c \approx 3.355$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you compute cumulative probabilities for the t-distribution?
2. What is the difference between one-tailed and two-tailed tests in t-distributions?
3. How does the number of degrees of freedom affect the shape of the t-distribution?
4. Can you explain why we subtract from 1 to find $P(t \geq x)$?
5. How do we find critical values using a t-distribution table?

Tip: When solving t-distribution problems, always check whether you're dealing with one-tailed or two-tailed probabilities, as this changes the critical value calculation.