Answer :
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.
