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### (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.

Consider a t-distribution with 10 degrees of freedom. Compute P(-1.64 < t < 1.64). Consider a t-distribution with 2 degrees of freedom. Find the value of c such that P(t ≥ c) = 0.05.

Learn how to solve t-distribution problems, including computing P(-1.64 < t < 1.64) for 10 degrees of freedom and finding the critical value c where P(t ≥ c) = 0.05 for 2 degrees of freedom. This solution involves key statistical concepts such as cumulative distribution functions and critical values.

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