a) Consider at distribution with 24 degrees of freedom. Compute P(t <- 1.51) , Round your answer to at least three decimal places.

P(t <= - 1.51) = 

(b) Consider at distribution with 5 degrees of freedom. Find the value of a such that P(- c < t < c) = 0.90 Round your answer to at least three decimal places.

c =

### Part (a)
You need to calculate the probability \( P(t \leq -1.51) \) for a t-distribution with 24 degrees of freedom.

For a t-distribution with 24 degrees of freedom, you can use the cumulative distribution function (CDF) to find this probability.

\[ P(t \leq -1.51) \approx 0.071 \]

### Part (b)
You need to find the value of \( c \) such that \( P(-c < t < c) = 0.90 \) for a t-distribution with 5 degrees of freedom.

This means you are looking for the t-value where the middle 90% of the distribution lies.

For a t-distribution with 5 degrees of freedom, the critical value \( c \) can be found using the t-table or a statistical calculator. The value of \( c \) is:

\[ c \approx 2.015 \]

### Summary:
- **(a)** \( P(t \leq -1.51) \approx 0.071 \)
- **(b)** \( c \approx 2.015 \)

Would you like more details or have any questions about these calculations? Here are some related questions:

1. How do degrees of freedom affect the shape of a t-distribution?
2. What is the difference between a t-distribution and a normal distribution?
3. How do you interpret the result from part (a) in a practical context?
4. Can you explain the concept of the confidence interval related to part (b)?
5. How would the value of \( c \) change if the confidence level was 95% instead of 90%?

**Tip:** The t-distribution is particularly useful when dealing with small sample sizes, as it accounts for the extra variability introduced by estimating the population standard deviation.

Learn how to calculate probabilities and critical values in t-distributions with 24 and 5 degrees of freedom. Includes step-by-step solutions for P(t <= -1.51) and finding the value of c such that P(-c < t < c) = 0.90. Suitable for advanced undergraduate students and anyone interested in statistical distributions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation