Let me help with the problems shown in your image.

(a) Consider a t-distribution with 10 degrees of freedom. Compute $P(-1.64 < t < 1.64)$.

We need to find the probability that a t-distributed random variable with 10 degrees of freedom lies between -1.64 and 1.64. This requires looking up the cumulative distribution function (CDF) of the t-distribution for 10 degrees of freedom.

Steps:

1. Find the cumulative probability for $t = -1.64$ and $t = 1.64$ using the t-distribution table or a calculator for the t-distribution.
2. Subtract the lower cumulative probability from the higher one to get the probability that the t-value lies between -1.64 and 1.64.

(b) Consider a t-distribution with 2 degrees of freedom. Find the value of $c$ such that $P(t \geq c) = 0.05$.

In this case, we're looking for the critical value $c$ where the upper tail probability (to the right of $c$) is 0.05. For a t-distribution with 2 degrees of freedom, we can:

1. Use a t-distribution table to find the t-value corresponding to a cumulative probability of $1 - 0.05 = 0.95$ for 2 degrees of freedom.
2. Alternatively, use a statistical calculator to find this critical t-value.

I will calculate both values for you now.Here are the results for the problems:

(a) For a t-distribution with 10 degrees of freedom:

$P(-1.64 < t < 1.64) = 0.868$ (rounded to three decimal places).

(b) For a t-distribution with 2 degrees of freedom:

The critical value $c$ such that $P(t \geq c) = 0.05$ is $c = 2.920$ (rounded to three decimal places).

Would you like further details or have any questions? Here are some related questions you could explore:

1. How do you compute probabilities using the cumulative distribution function (CDF) of a t-distribution?
2. What is the difference between one-tailed and two-tailed t-tests?
3. How does the degrees of freedom affect the shape of a t-distribution?
4. Can you explain why we subtract the two cumulative probabilities in part (a)?
5. How do you use the inverse t-distribution to find critical values?

Tip: When working with t-distributions, always double-check the degrees of freedom, as they significantly affect your results.