## How to Construct a Confidence Interval From Data Using the t-Distribution

You may be asked in elementary statistics to construct a confidence interval from a given set of data. Thanks to the t-distribution table, only a few short steps are needed to calculate the t-distribution.

Sample problem: Construct a 98% Confidence Interval based on the following data: 45, 55, 67, 45, 68, 79, 98, 87, 84, 82

Step 1: Find the mean and standard deviation for the data.
Mean = 71 (Click here if you want to know how to find the mean)
Standard Deviation: 18.172. (Click here if you want to know how to find the standard deviation)

Step 2:Subtract 1 from your sample size to find the degrees of freedom (df). We have 10 numbers listed, so our sample size is 10, so our df = 9. You’ll need this number in step 4.

Step 3: Subtract the confidence level from 1, then divide by two.
(1 – .98) / 2 = .01

Step 4:Look up df (Step 2) and α (Step 3) in the t-distribution table. For df = 9 and α = .01, the table gives us 2.821.

Step 4:Divide your  standard deviation (step 1) by the square root of your sample size.
18.172 / √(10) = 5.75

Step 5:: Multiply step 3 by step 4.
2.821 × 5.75 = 16.22075
Step 6: For the lower end of the range, subtract step 5 from the mean (Step 1).
71 – 16.22075 = 54.77925

Step 7:For the upper end of the range, add step 5 to the  mean (Step 1).
71 + 16.22075 = 87.22075

That’s it! Like the explanation? Check out our statistics how-to book, with a how-to for every elementary statistics problem type.

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## 18 Responses to “How to Construct a Confidence Interval From Data Using the t-Distribution”

1. ### Angie Widdows said:

Oct 08, 09 at 8:23 am

This example was a little hard to understand. It would be helpful in step 1 to show how the mean was found and where the standard deviation number came from. I understand where the numbers came from but someone else might not. Also, in step 2, is there a logical reason why 1 is subtracted from the degrees of freedom?

2. ### Stephanie said:

Oct 09, 09 at 5:58 am

i Angie.

1 is subtracted from n to get the degrees of freedom. There is indeed a logical reason for n-1 equaling df, but it’s beyond the scope of this course unfortunately.

If you really want to know, I think this article explains it well:
http://www.jerrydallal.com/LHSP/dof.htm

Stephanie

3. ### Jennifer Thomas said:

Oct 10, 09 at 7:30 am

This was extremely helpful. When using the show me tool in mathzone, it did not provide step by step instructions on how to find the confidence interval.

Thank you!!

Jennifer Thomas

4. ### Rebecca Gamble said:

Oct 11, 09 at 8:49 pm

This and the show me or guide me is a good tool to use when doing the homework. The computer breaks the problem down and guides you through solving the problem.

5. ### Shannon Manns said:

Oct 17, 09 at 9:34 pm

Thank goodness for the calculator explaining the variance and standard deviation. It was very helpful to working out the problems.

6. ### Donna Allen said:

Oct 24, 09 at 2:36 pm

This example was very helpful to me. And, I really appreciated the refresher on calculating the mean and standard deviation. Thanks!

7. ### April Fulton said:

Oct 25, 09 at 8:42 pm

This step by step example really explained to me what I was not doing in my problems. Now I can refer back to the step by step process to help me in my work.

8. ### Lisa Barcomb said:

Nov 01, 09 at 8:23 pm

This I understood because it broke it down step by step and made you see how the problem worked so you could get the right answer. This problem was real busy it had a few steps to it as well. And it seems like if you don’t do all the steps your problem is wrong. But for the most part this was very helpful.

9. ### Joni Poore said:

Mar 07, 10 at 7:41 pm

Again, extrememly helpful unlike the book.
I liked that you added in the mean and standard deviation, sadly I forgot how to do SD without sample size!

10. ### Stephanie said:

Mar 22, 10 at 10:27 am

Thanks for spotting the error Megan, I appreciate it and I’m sure future students will also!

11. ### Alison Bryant said:

Mar 22, 10 at 10:34 am

I love how this is set up step by step I was really helpful and clarifying. I wrote down all the steps like I did with the confidence and width problems and now I have almost memorized the procedure, thanks this was again very, very helpful.

12. ### Zen said:

Dec 14, 11 at 11:20 am

Hi Stephanie,
This is a great example! However my textbook is confusing me in that it says df=n and the T-table it provides is also different from the one here. Instead of having “a” ranging from 0.1 to 0.0005, it has “p” ranging from 0.6 to 0.9995

Why is that?

13. ### Andale said:

Mar 04, 12 at 5:33 pm

I’d have to view the actual text to tell you. I have no idea why df would equal n. Send me a scan of the relevant pages to andalepublishing at gmail and I’ll take a look.

Stephanie

14. ### Erix said:

May 31, 12 at 12:06 pm

Thank you for this example, it was so easy to understand. Keep up the good work.

Thanks!

15. ### Mandy said:

Jul 08, 12 at 9:16 am

What if the problem doesn’t have a standard deviation?

16. ### Andale said:

Jul 22, 12 at 8:14 am

You should be able to calculate the standard deviation from the sample data given. I take it you don’t have sample data? If you could post the actual problem from the text or homework, I will take a look.

Stephanie

17. ### Taylor Conley said:

Mar 01, 13 at 7:04 pm

This was very helpful. I have to construct a confidence interval for my stats and research methods class and this was very clear. Thank you!

18. ### Andale said:

Mar 04, 13 at 5:59 pm