Let's solve the problems step by step:

(a) Compute $P(t \leq -2)$ for a t-distribution with 25 degrees of freedom.

We use the cumulative distribution function (CDF) for the t-distribution. To find $P(t \leq -2)$, we can directly compute this using statistical software or a t-distribution table.

Steps:

1. Use a statistical table or software to find the CDF at $t = -2$ for $df = 25$.
2. $P(t \leq -2)$ is the area to the left of $t = -2$.

Calculation using statistical software: $P(t \leq -2)$ = 0.0297 (approximately).

(b) Find the value of $c$ such that $P(-c < t < c) = 0.90$ for a t-distribution with 20 degrees of freedom.

Explanation:

1. The probability $P(-c < t < c) = 0.90$ implies that 90% of the distribution lies between $-c$ and $c$, leaving 5% in each tail.
2. This corresponds to finding the critical t-value where $P(t < -c) = 0.05$ or $P(t > c) = 0.05$.
3. Using a t-distribution table or statistical software for $df = 20$, find the value of $c$ such that the central 90% is within $-c$ and $c$.

Calculation: For $df = 20$, the critical value $c$ is approximately: $c = 1.7247$.

Let me know if you would like detailed explanations for any steps or assistance in verifying these results!

5 Related Questions:

1. How is the cumulative distribution function (CDF) used in finding probabilities for t-distributions?
2. How does the number of degrees of freedom influence the shape of the t-distribution?
3. Can you explain the difference between one-tailed and two-tailed probabilities in hypothesis testing?
4. What is the significance of the 90% confidence interval in part (b)?
5. How do you use software tools like Python or Excel to compute t-distribution probabilities?

Tip:

Always check the degrees of freedom and ensure whether the problem requires cumulative probabilities or critical values to avoid confusion.