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Use the ALEKS calculator to solve the following problems.

(a) Consider a [tex]$t$[/tex] distribution with 25 degrees of freedom. Compute [tex]$P(t \geq 1.72)$[/tex]. Round your answer to at least three decimal places.

[tex]\[ P(t \geq 1.72) = \square \][/tex]

(b) Consider a [tex]$t$[/tex] distribution with 8 degrees of freedom. Find the value of [tex]$c$[/tex] such that [tex]$P(-c \ \textless \ t \ \textless \ c) = 0.95$[/tex]. Round your answer to at least three decimal places.

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

Sagot :

GinnyAnsweridn
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]