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Sagot :
Absolutely, let's go through these problems step-by-step.
### Part (a)
We are given a [tex]\( t \)[/tex]-distribution with 25 degrees of freedom and asked to compute [tex]\( P(t \geq 1.72) \)[/tex].
1. Identify the distribution: We are working with a [tex]\( t \)[/tex]-distribution with 25 degrees of freedom.
2. Locate the value on the [tex]\( t \)[/tex]-distribution: We need to find the probability that the [tex]\( t \)[/tex]-value is greater than or equal to 1.72.
3. Find the cumulative distribution function (CDF) value for [tex]\( t = 1.72 \)[/tex]:
This gives the area under the curve to the left of [tex]\( t = 1.72 \)[/tex]. Let's denote the CDF value as [tex]\( P(t \leq 1.72) \)[/tex].
4. Compute the desired probability:
Since the total area under the curve is 1, [tex]\( P(t \geq 1.72) \)[/tex] is the complement of [tex]\( P(t \leq 1.72) \)[/tex]:
[tex]\[ P(t \geq 1.72) = 1 - P(t \leq 1.72) \][/tex]
After performing these calculations, we find:
[tex]\[ P(t \geq 1.72) = 0.049 \][/tex]
### Part (b)
We are given a [tex]\( t \)[/tex]-distribution with 8 degrees of freedom and asked to find the value of [tex]\( c \)[/tex] such that [tex]\( P(-c < t < c) = 0.95 \)[/tex].
1. Identify the distribution: We are working with a [tex]\( t \)[/tex]-distribution with 8 degrees of freedom.
2. Understand the probability requirement:
[tex]\( P(-c < t < c) = 0.95 \)[/tex] implies that we want the area in the middle of the distribution to be 0.95.
Since the [tex]\( t \)[/tex]-distribution is symmetric, this leaves 0.025 (i.e., 2.5%) in each tail.
3. Find the critical value [tex]\( c \)[/tex]:
We need to determine the critical [tex]\( t \)[/tex]-value such that the area to the left of [tex]\( c \)[/tex] is 0.975 (0.95 + 0.025), as [tex]\( P(-c < t < c) = 0.95 \)[/tex] includes both tails.
We'll look up or calculate the value for which the CDF is 0.975 with 8 degrees of freedom.
This critical [tex]\( t \)[/tex]-value is found to be:
[tex]\[ c = 2.306 \][/tex]
So, we have computed:
[tex]\[ (a) \, P(t \geq 1.72) = 0.049 \][/tex]
[tex]\[ (b) \, c = 2.306 \][/tex]
### Part (a)
We are given a [tex]\( t \)[/tex]-distribution with 25 degrees of freedom and asked to compute [tex]\( P(t \geq 1.72) \)[/tex].
1. Identify the distribution: We are working with a [tex]\( t \)[/tex]-distribution with 25 degrees of freedom.
2. Locate the value on the [tex]\( t \)[/tex]-distribution: We need to find the probability that the [tex]\( t \)[/tex]-value is greater than or equal to 1.72.
3. Find the cumulative distribution function (CDF) value for [tex]\( t = 1.72 \)[/tex]:
This gives the area under the curve to the left of [tex]\( t = 1.72 \)[/tex]. Let's denote the CDF value as [tex]\( P(t \leq 1.72) \)[/tex].
4. Compute the desired probability:
Since the total area under the curve is 1, [tex]\( P(t \geq 1.72) \)[/tex] is the complement of [tex]\( P(t \leq 1.72) \)[/tex]:
[tex]\[ P(t \geq 1.72) = 1 - P(t \leq 1.72) \][/tex]
After performing these calculations, we find:
[tex]\[ P(t \geq 1.72) = 0.049 \][/tex]
### Part (b)
We are given a [tex]\( t \)[/tex]-distribution with 8 degrees of freedom and asked to find the value of [tex]\( c \)[/tex] such that [tex]\( P(-c < t < c) = 0.95 \)[/tex].
1. Identify the distribution: We are working with a [tex]\( t \)[/tex]-distribution with 8 degrees of freedom.
2. Understand the probability requirement:
[tex]\( P(-c < t < c) = 0.95 \)[/tex] implies that we want the area in the middle of the distribution to be 0.95.
Since the [tex]\( t \)[/tex]-distribution is symmetric, this leaves 0.025 (i.e., 2.5%) in each tail.
3. Find the critical value [tex]\( c \)[/tex]:
We need to determine the critical [tex]\( t \)[/tex]-value such that the area to the left of [tex]\( c \)[/tex] is 0.975 (0.95 + 0.025), as [tex]\( P(-c < t < c) = 0.95 \)[/tex] includes both tails.
We'll look up or calculate the value for which the CDF is 0.975 with 8 degrees of freedom.
This critical [tex]\( t \)[/tex]-value is found to be:
[tex]\[ c = 2.306 \][/tex]
So, we have computed:
[tex]\[ (a) \, P(t \geq 1.72) = 0.049 \][/tex]
[tex]\[ (b) \, c = 2.306 \][/tex]
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