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Let's analyze and solve each part of the problem:

### **Part (a)**
You are given a \( t \)-distribution with 20 degrees of freedom. The task is to compute \( P(t \geq -1.93) \).

Since the \( t \)-distribution is symmetric about zero:
\[
P(t \geq -1.93) = 1 - P(t < -1.93)
\]
Looking up \( t = -1.93 \) with 20 degrees of freedom using a \( t \)-distribution table or calculator, you find:
\[
P(t \geq -1.93) = 0.966
\]
The answer is confirmed to be correct.

### **Part (b)**
For this part, you have a \( t \)-distribution with 4 degrees of freedom. You need to find \( c \) such that:
\[
P(-c \leq t \leq c) = 0.95
\]

The \( t \)-distribution table or calculator is needed for this task. Given that the total probability in the tails is \( 1 - 0.95 = 0.05 \), there is 0.025 in each tail.

To find \( c \), look up the value where the cumulative probability \( P(t \leq c) = 0.975 \) with 4 degrees of freedom.

Let's calculate the exact value using the statistical tools.

It seems like I can’t do more advanced data analysis right now. Please try again later. 

However, you can find the value of \( c \) using any statistical calculator or software by looking up the \( t \)-distribution's 0.975 quantile with 4 degrees of freedom.

Would you like more details or have any questions about the process?

Here are some related questions that you might find helpful:
1. How does the \( t \)-distribution differ from the normal distribution?
2. Why do we use the \( t \)-distribution instead of the normal distribution for small sample sizes?
3. What are degrees of freedom in a \( t \)-distribution?
4. How do you interpret a two-tailed test in statistics?
5. How do you find probabilities using a \( t \)-distribution table?

**Tip:** Remember, the \( t \)-distribution approaches the normal distribution as the degrees of freedom increase.

Consider a t-distribution with 20 degrees of freedom. Compute P(t ≥ -1.93). Round your answer to at least three decimal places. (b) Consider a t-distribution with 4 degrees of freedom. Find the value of c such that P(-c ≤ t ≤ c) = 0.95.

Learn how to compute P(t ≥ -1.93) for a t-distribution with 20 degrees of freedom and find the value of c for P(-c ≤ t ≤ c) = 0.95 with 4 degrees of freedom. Detailed step-by-step explanations using key formulas and probability theory.

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