Certainly! Let’s go through the calculations step-by-step:
### Part (a)
We are asked to compute the probability that a random variable with a [tex]\( t \)[/tex]-distribution with 26 degrees of freedom lies between [tex]\(-1.38\)[/tex] and [tex]\(1.38\)[/tex].
1. Identify the cumulative probabilities:
- For a [tex]\( t \)[/tex]-distribution with 26 degrees of freedom, calculate the cumulative distribution function (CDF) values at [tex]\(-1.38\)[/tex] and [tex]\(1.38\)[/tex].
- Let's denote these values as [tex]\(P(t \leq -1.38)\)[/tex] and [tex]\(P(t \leq 1.38)\)[/tex].
2. Use the cumulative probabilities to find the target probability:
- The probability we are interested in is given by subtracting these two cumulative probabilities:
[tex]\[
P(-1.38 < t < 1.38) = P(t \leq 1.38) - P(t \leq -1.38)
\][/tex]
3. Calculate the values:
- After calculating the above values, we will find that:
[tex]\[
P(-1.38 < t < 1.38) \approx 0.821
\][/tex]
So, the probability [tex]\(P(-1.38 < t < 1.38)\)[/tex] for a [tex]\( t \)[/tex]-distribution with 26 degrees of freedom, rounded to three decimal places, is:
[tex]\[
P(-1.38 < t < 1.38) = 0.821
\][/tex]
### Part (b)
We need to find the value of [tex]\(c\)[/tex] such that [tex]\( P(t \leq c) = 0.05 \)[/tex] for a [tex]\( t \)[/tex]-distribution with 21 degrees of freedom.
1. Identify the quantile:
- For a [tex]\( t \)[/tex]-distribution with 21 degrees of freedom, find the [tex]\(0.05\)[/tex] quantile. This is the value of [tex]\(c\)[/tex] such that the cumulative distribution function value at [tex]\(c\)[/tex] is [tex]\(0.05\)[/tex].
2. Determine the corresponding [tex]\( t \)[/tex]-value:
- Using the quantile function (inverse of the CDF), we find that:
[tex]\[
P(t \leq c) = 0.05 \implies c \approx -1.721
\][/tex]
So, the value of [tex]\(c\)[/tex] such that [tex]\( P(t \leq c) = 0.05 \)[/tex] for a [tex]\( t \)[/tex]-distribution with 21 degrees of freedom, rounded to three decimal places, is:
[tex]\[
c = -1.721
\][/tex]
In summary:
(a) [tex]\(P(-1.38 < t < 1.38) = 0.821\)[/tex]
(b) [tex]\(c = -1.721\)[/tex]
