https://file.aiquickdraw.com/mathbot/file/1727981689980prqg7b86.jpg

The image contains two problems related to the **t-distribution**. Here is the breakdown of each problem and how to solve them:

### (a) Problem:
- **Consider a t-distribution with 20 degrees of freedom**. Compute the probability:

\[
P(-1.58 < t < 1.58)
\]

The goal is to find the probability that the t-statistic lies between -1.58 and 1.58, with 20 degrees of freedom. You can use a **t-distribution calculator** or look up the cumulative probability from a **t-distribution table**.

Steps to solve:
1. Find the cumulative probability for **t = 1.58** for **20 degrees of freedom (df = 20)**.
2. Because the t-distribution is symmetric, the probability for **P(-1.58 < t < 1.58)** can be found by doubling the cumulative probability for **t = 1.58** and subtracting from 1.

The formula is:
\[
P(-1.58 < t < 1.58) = 2 \times P(t < 1.58) - 1
\]

### (b) Problem:
- **Consider a t-distribution with 6 degrees of freedom**. Find the value of **c** such that:

\[
P(t \leq c) = 0.01
\]

Here, you are looking for the critical value of **c** for which the cumulative probability is **0.01** for a **t-distribution with 6 degrees of freedom (df = 6)**. This is a lower-tail problem, so you would look up the **inverse cumulative probability** for **0.01** in a **t-distribution table** with **6 degrees of freedom** or use a calculator.

Let me know if you would like further help solving these problems, or if you need more details!

---

**Would you like the detailed step-by-step solution for these problems?**

Here are some related questions for further practice:
1. How do you compute the cumulative probability from a t-distribution?
2. What is the difference between a t-distribution and a normal distribution?
3. How can you use a t-table to find probabilities?
4. What is the effect of degrees of freedom on the t-distribution?
5. How do you interpret a p-value in the context of hypothesis testing with t-distribution?

**Tip:** In hypothesis testing, using a t-distribution is crucial when working with small sample sizes (typically n < 30) or when the population standard deviation is unknown.

Consider a t-distribution with 20 degrees of freedom. Compute P(-1.58 < t < 1.58). Round your answer to at least three decimal places.
Consider a t-distribution with 6 degrees of freedom. Find the value of c such that P(t ≤ c) = 0.01. Round your answer to at least three decimal places.

Learn how to compute the probability for a t-distribution with 20 degrees of freedom and solve for a critical value for a t-distribution with 6 degrees of freedom. Detailed explanation with cumulative probability methods and inverse lookup for t-distributions.

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