(a)Consider a *t* distribution with 5 degrees of freedom. Compute P(-1.61 less than t less than 1.61). Round your answer to at least three decimal places.

To calculate the probability \( P(-1.61 < t < 1.61) \) for a t-distribution with 5 degrees of freedom, you can follow these steps:

1. **Determine the cumulative probability** using the t-distribution table or a calculator.
   - Find \( P(t < 1.61) \) for \( t \) with 5 degrees of freedom.
   - Find \( P(t < -1.61) \) for \( t \) with 5 degrees of freedom. Since the t-distribution is symmetric, \( P(t < -1.61) = 1 - P(t < 1.61) \).

2. **Calculate the probability** \( P(-1.61 < t < 1.61) \):
   \[
   P(-1.61 < t < 1.61) = P(t < 1.61) - P(t < -1.61)
   \]

Let's compute this using the cumulative distribution function (CDF) for the t-distribution.

The CDF for \( t = 1.61 \) and 5 degrees of freedom gives us \( P(t < 1.61) \).

I'll calculate this now.The probability \( P(-1.61 < t < 1.61) \) for a t-distribution with 5 degrees of freedom is approximately **0.832**.

Do you want further details or have any questions?

Here are some related questions you might be interested in:
1. What is the t-distribution and how is it used in statistics?
2. How does the degrees of freedom affect the shape of the t-distribution?
3. What are some common applications of the t-distribution in hypothesis testing?
4. How can we use a t-distribution table to find probabilities for different t-values?
5. What is the difference between a t-distribution and a normal distribution?
6. How do we interpret the result of a t-test in hypothesis testing?
7. What are the assumptions required for a t-test to be valid?
8. How would the probability change if the degrees of freedom were increased to 10?

**Tip:** When working with t-distributions, remember that they approach the normal distribution as the degrees of freedom increase.

Learn how to calculate the probability P(-1.61 < t < 1.61) for a T-distribution with 5 degrees of freedom. Detailed steps using cumulative distribution functions (CDF) are provided, ensuring accurate results rounded to three decimal places.

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