https://file.aiquickdraw.com/mathbot/file/1731096051602ntzf4rni.jpg

Let's solve each part step-by-step.

### Part (a)

We need to calculate the probability for a \( t \)-distribution with 7 degrees of freedom:
\[
P(-1.17 < t < 1.17)
\]

This probability can be found using a \( t \)-distribution table or software. The steps involve:

1. Finding \( P(t < 1.17) \) with 7 degrees of freedom.
2. Finding \( P(t < -1.17) \) with 7 degrees of freedom.
3. Subtracting these probabilities to find the value of \( P(-1.17 < t < 1.17) \).

Using a calculator or statistical software, we find:

\[
P(t < 1.17) \approx 0.847
\]
\[
P(t < -1.17) \approx 0.153
\]

Thus,
\[
P(-1.17 < t < 1.17) = 0.847 - 0.153 = 0.694
\]

So, the answer for part (a) is approximately:
\[
\boxed{0.694}
\]

### Part (b)

For part (b), we need to find the value of \( c \) for a \( t \)-distribution with 16 degrees of freedom such that:
\[
P(t \geq c) = 0.10
\]

This means that \( c \) is the 90th percentile for a \( t \)-distribution with 16 degrees of freedom. We can find this by looking up the critical value in a \( t \)-table or using software.

From a \( t \)-table or statistical calculator, we find:
\[
c \approx 1.337
\]

So, the answer for part (b) is approximately:
\[
\boxed{1.337}
\]

---

Would you like further details or have any other questions? 

Here are five related questions to deepen your understanding:

1. How does the degrees of freedom affect the shape of the \( t \)-distribution?
2. What is the interpretation of a two-tailed probability in the context of a \( t \)-distribution?
3. How can you use software like Python or R to calculate probabilities for the \( t \)-distribution?
4. What is the relationship between \( t \)-distribution and the normal distribution as degrees of freedom increase?
5. How can you determine critical values for a one-tailed vs. two-tailed test using the \( t \)-distribution?

**Tip:** When working with probability distributions, using a software or calculator is efficient for finding precise values, especially with non-standard critical points.

Consider a t distribution with 7 degrees of freedom. Compute P(-1.17 < t < 1.17). Round your answer to at least three decimal places.

Consider a t distribution with 16 degrees of freedom. Find the value of c such that P(t ≥ c) = 0.10. Round your answer to at least three decimal places.

Learn how to compute probabilities and find critical values in a t-distribution. This solution shows step-by-step calculations for P(-1.17 < t < 1.17) with 7 degrees of freedom and determines the 90th percentile for a t-distribution with 16 degrees of freedom. Perfect for college-level introductory statistics students.

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