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.