Welcome to Westonci.ca, where your questions are met with accurate answers from a community of experts and enthusiasts. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Let's solve these problems step-by-step.
### Part (a): Compute \( P(t \geq -1.2) \) for a \( t \)-distribution with 24 degrees of freedom.
1. Understand the Problem: We need to find the probability that a \( t \)-value is greater than or equal to \(-1.2\) given that the \( t \)-distribution has 24 degrees of freedom.
2. Find the Cumulative Distribution Function (CDF): The CDF provides the probability that the \( t \)-value will be less than or equal to a certain value. For a \( t \)-distribution with 24 degrees of freedom:
[tex]\[ P(t \leq -1.2) \][/tex]
3. Calculate \( P(t \geq -1.2) \): The value we need is the complement of the CDF value at -1.2. Therefore:
[tex]\[ P(t \geq -1.2) = 1 - P(t \leq -1.2) \][/tex]
4. Result: After calculating or using statistical tables/software:
[tex]\[ P(t \geq -1.2) \approx 0.879 \][/tex]
### Part (b): Find the value of \( c \) for a \( t \)-distribution with 20 degrees of freedom such that \( P(-c < t < c) = 0.90 \).
1. Understand the Problem: We need to find the critical value \( c \) such that the area under the \( t \)-distribution curve between \(-c\) and \( c \) covers 90% of the total probability. This implies:
[tex]\[ P(-c < t < c) = 0.90 \][/tex]
2. Symmetry and Total Probability: The total area under the \( t \)-distribution curve is 1. The probability outside the interval \([-c, c]\) is \( 0.10 \), so:
[tex]\[ P(t < -c) + P(t > c) = 0.10 \][/tex]
Given symmetry,
[tex]\[ P(t < -c) = P(t > c) = 0.05 \][/tex]
3. Use Inverse CDF Function:
The value of \( c \) can be found by using the inverse CDF (or quantile function) for the \( t \)-distribution. We need the \( t \)-value corresponding to the cumulative probability of \( 0.95 \) (as the remaining \( 0.05 \) is in the upper tail).
4. Result: For 20 degrees of freedom:
[tex]\[ c \approx 1.725 \][/tex]
### Summary:
- Part (a): \( P(t \geq -1.2) \approx 0.879 \)
- Part (b): \( c \approx 1.725 \)
These values provide the solutions to the problems as asked.
### Part (a): Compute \( P(t \geq -1.2) \) for a \( t \)-distribution with 24 degrees of freedom.
1. Understand the Problem: We need to find the probability that a \( t \)-value is greater than or equal to \(-1.2\) given that the \( t \)-distribution has 24 degrees of freedom.
2. Find the Cumulative Distribution Function (CDF): The CDF provides the probability that the \( t \)-value will be less than or equal to a certain value. For a \( t \)-distribution with 24 degrees of freedom:
[tex]\[ P(t \leq -1.2) \][/tex]
3. Calculate \( P(t \geq -1.2) \): The value we need is the complement of the CDF value at -1.2. Therefore:
[tex]\[ P(t \geq -1.2) = 1 - P(t \leq -1.2) \][/tex]
4. Result: After calculating or using statistical tables/software:
[tex]\[ P(t \geq -1.2) \approx 0.879 \][/tex]
### Part (b): Find the value of \( c \) for a \( t \)-distribution with 20 degrees of freedom such that \( P(-c < t < c) = 0.90 \).
1. Understand the Problem: We need to find the critical value \( c \) such that the area under the \( t \)-distribution curve between \(-c\) and \( c \) covers 90% of the total probability. This implies:
[tex]\[ P(-c < t < c) = 0.90 \][/tex]
2. Symmetry and Total Probability: The total area under the \( t \)-distribution curve is 1. The probability outside the interval \([-c, c]\) is \( 0.10 \), so:
[tex]\[ P(t < -c) + P(t > c) = 0.10 \][/tex]
Given symmetry,
[tex]\[ P(t < -c) = P(t > c) = 0.05 \][/tex]
3. Use Inverse CDF Function:
The value of \( c \) can be found by using the inverse CDF (or quantile function) for the \( t \)-distribution. We need the \( t \)-value corresponding to the cumulative probability of \( 0.95 \) (as the remaining \( 0.05 \) is in the upper tail).
4. Result: For 20 degrees of freedom:
[tex]\[ c \approx 1.725 \][/tex]
### Summary:
- Part (a): \( P(t \geq -1.2) \approx 0.879 \)
- Part (b): \( c \approx 1.725 \)
These values provide the solutions to the problems as asked.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.