Answered

IDNLearn.com provides a comprehensive platform for finding accurate answers. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.

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]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.