Math / Data & Statistics

QuestionUse a t-distribution to answer this question. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a $t$ -statistic will be used for inference about the difference in sample means. State the degrees of freedom used.
Find the endpoints of the $t$ -distribution with $5 \%$ beyond them in each tail if the samples have sizes $n_{1}=8$ and $n_{2}=10$ . Enter the exact number for the degrees of freedom and round your answer for the endpoints to two decimal places. degrees of freedom = $\square$ endpoints $= \pm \mathbf{i}$

Studdy Solution

STEP 1

What is this asking? We need to find the degrees of freedom and the critical t-value for a two-tailed t-distribution with a combined sample size of 18 and a significance level of 0.10 (that's 5% in each tail). Watch out! Don't mix up one-tailed and two-tailed t-values!
We're looking for the t-value that cuts off 5% in each tail, not a total of 5%.

STEP 2

1. Calculate Degrees of Freedom
2. Find the Critical t-Value

STEP 3

When dealing with two samples, the degrees of freedom for a t-distribution are calculated using a slightly tricky formula.
It's not just adding the sample sizes!
The formula is: $df = n_1 + n_2 - 2$ , where $n_1$ is the size of the first sample and $n_2$ is the size of the second sample.

STEP 4

In our case, $n_1 = \mathbf{8}$ and $n_2 = \mathbf{10}$ .
Let's plug these values into our formula: $df = \mathbf{8} + \mathbf{10} - 2$ .

STEP 5

So, $df = \mathbf{18} - 2 = \mathbf{16}$ .
We have 16 degrees of freedom!

STEP 6

We're looking for the t-value that leaves 5% in each tail.
This means we're looking for a two-tailed t-value with an alpha of 10% (5% in each tail adds up to 10%).
We also know our degrees of freedom are 16.

STEP 7

Now, we need to consult a t-table or use a calculator to find this critical t-value.
Look up the value corresponding to 16 degrees of freedom and a two-tailed alpha of 0.10.

STEP 8

The critical t-value is approximately $\pm \mathbf{1.746}$ .
This means that 5% of the t-distribution lies beyond $\mathbf{1.746}$ and another 5% lies below $-\mathbf{1.746}$ .

STEP 9

Degrees of freedom = 16 Endpoints = $\pm \mathbf{1.746}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate Degrees of Freedom
2. Find the Critical t-Value

STEP 3

When dealing with two samples, the degrees of freedom for a t-distribution are calculated using a slightly tricky formula.
It's not just adding the sample sizes!
The formula is: $df = n_1 + n_2 - 2$ , where $n_1$ is the size of the first sample and $n_2$ is the size of the second sample.

STEP 4

In our case, $n_1 = \mathbf{8}$ and $n_2 = \mathbf{10}$ .
Let's plug these values into our formula: $df = \mathbf{8} + \mathbf{10} - 2$ .

STEP 5

So, $df = \mathbf{18} - 2 = \mathbf{16}$ .
We have 16 degrees of freedom!

STEP 6

We're looking for the t-value that leaves 5% in each tail.
This means we're looking for a two-tailed t-value with an alpha of 10% (5% in each tail adds up to 10%).
We also know our degrees of freedom are 16.

STEP 7

Now, we need to consult a t-table or use a calculator to find this critical t-value.
Look up the value corresponding to 16 degrees of freedom and a two-tailed alpha of 0.10.

STEP 8

The critical t-value is approximately $\pm \mathbf{1.746}$ .
This means that 5% of the t-distribution lies beyond $\mathbf{1.746}$ and another 5% lies below $-\mathbf{1.746}$ .

STEP 9

Degrees of freedom = 16 Endpoints = $\pm \mathbf{1.746}$

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Studdy Solution

STEP 1

STEP 2

1. Calculate Degrees of Freedom2. Find the Critical t-Value

STEP 3

STEP 4

In our case, n1=8n_1 = \mathbf{8}n1​=8 and n2=10n_2 = \mathbf{10}n2​=10.Let's plug these values into our formula: df=8+10−2df = \mathbf{8} + \mathbf{10} - 2df=8+10−2.

STEP 5

So, df=18−2=16df = \mathbf{18} - 2 = \mathbf{16}df=18−2=16.We have **16** degrees of freedom!

STEP 6

We're looking for the t-value that leaves 5% in each tail.This means we're looking for a two-tailed t-value with an *alpha* of 10% (5% in each tail adds up to 10%).We also know our degrees of freedom are **16**.

STEP 7

Now, we need to consult a t-table or use a calculator to find this critical t-value.Look up the value corresponding to **16** degrees of freedom and a two-tailed alpha of **0.10**.

STEP 8

The critical t-value is approximately ±1.746\pm \mathbf{1.746}±1.746.This means that 5% of the t-distribution lies beyond 1.746\mathbf{1.746}1.746 and another 5% lies below −1.746-\mathbf{1.746}−1.746.

STEP 9

Degrees of freedom = **16** Endpoints = ±1.746\pm \mathbf{1.746}±1.746

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1. Calculate Degrees of Freedom
2. Find the Critical t-Value

In our case, $n_1 = \mathbf{8}$ and $n_2 = \mathbf{10}$ .
Let's plug these values into our formula: $df = \mathbf{8} + \mathbf{10} - 2$ .

So, $df = \mathbf{18} - 2 = \mathbf{16}$ .
We have 16 degrees of freedom!

We're looking for the t-value that leaves 5% in each tail.
This means we're looking for a two-tailed t-value with an alpha of 10% (5% in each tail adds up to 10%).
We also know our degrees of freedom are 16.

Now, we need to consult a t-table or use a calculator to find this critical t-value.
Look up the value corresponding to 16 degrees of freedom and a two-tailed alpha of 0.10.

The critical t-value is approximately $\pm \mathbf{1.746}$ .
This means that 5% of the t-distribution lies beyond $\mathbf{1.746}$ and another 5% lies below $-\mathbf{1.746}$ .

Degrees of freedom = 16 Endpoints = $\pm \mathbf{1.746}$