IDNLearn.com is designed to help you find the answers you need quickly and easily. Join our knowledgeable community to find the answers you need for any topic or issue.

Use the ALEKS calculator to solve the following problems:

(a) Consider a [tex]t[/tex] distribution with 24 degrees of freedom. Compute [tex]P(t \geq -1.2)[/tex]. Round your answer to at least three decimal places.
[tex]\[
P(t \geq -1.2) = \square
\][/tex]

(b) Consider a [tex]t[/tex] distribution with 20 degrees of freedom. Find the value of [tex]c[/tex] such that [tex]P(-c \ \textless \ t \ \textless \ c) = 0.90[/tex]. Round your answer to at least three decimal places.
[tex]\[
c = \square
\][/tex]


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.