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(a) Consider a [tex]$t$[/tex] distribution with 26 degrees of freedom. Compute [tex]$P(-1.38\ \textless \ t\ \textless \ 1.38)$[/tex]. Round your answer to at least three decimal places.

[tex]\[ P(-1.38 \ \textless \ t \ \textless \ 1.38) = \ \square \][/tex]

(b) Consider a [tex]$t$[/tex] distribution with 21 degrees of freedom. Find the value of [tex]$c$[/tex] such that [tex]$P(t \leq c) = 0.05$[/tex]. Round your answer to at least three decimal places.

[tex]\[ c = \ \square \][/tex]

Sagot :

GinnyAnsweridn
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]
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