Math  /  Data & Statistics

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Current Attempt in Progress Is a t-Distribution Appropriate? A sample with size n=75n=75 has xˉ=18.92\bar{x}=18.92, and s=10.1s=10.1. The dotplot for this sample is given below.
Indicate whether or not it is appropriate to use the tt-distribution. \square If it is appropriate, give the degrees of freedom for the tt-distribution and give the estimated standard error. If it is not appropriate, enter -1 in both of the answer fields below.
Enter the exact answer for the degrees of freedom and round your answer for the standard error to two decimal places. df=d f= \square i standard error = \square i eTextbook and Media Save for Later Attempts: 0 of 5 used Submit Answer

Studdy Solution

STEP 1

What is this asking? Can we use a *t*-distribution with this sample's size, mean, and standard deviation, and if so, what are the degrees of freedom and standard error? Watch out! We need to check if the sample size is large enough or if the data is roughly normal to use the *t*-distribution.

STEP 2

1. Check for t-distribution appropriateness
2. Calculate degrees of freedom
3. Calculate standard error

STEP 3

The sample size is n=75n = 75.
Since 75>3075 > 30, the sample size is considered **large enough** to use the *t*-distribution, even if the population isn't perfectly normal!
Woohoo!

STEP 4

The degrees of freedom (dfdf) for a *t*-distribution are calculated as df=n1df = n - 1, where nn is the sample size.
In our case, n=75n = \mathbf{75}, so df=751=74df = 75 - 1 = \mathbf{74}.

STEP 5

The estimated standard error (SESE) is calculated using the sample standard deviation (ss) and the sample size (nn): SE=snSE = \frac{s}{\sqrt{n}}.

STEP 6

We have s=10.1s = \mathbf{10.1} and n=75n = \mathbf{75}.
So, SE=10.175SE = \frac{10.1}{\sqrt{75}}.

STEP 7

SE=10.17510.18.661.17SE = \frac{10.1}{\sqrt{75}} \approx \frac{10.1}{8.66} \approx \mathbf{1.17}.
Rounded to two decimal places, the standard error is **1.17**.

STEP 8

Yes, it's appropriate to use the *t*-distribution!
The degrees of freedom are 7474, and the estimated standard error is 1.171.17.

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