Statistics How To

Construct a Confidence Interval with the t-Distribution

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Construct a Confidence Interval : Overview

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.

construct a confidence interval

The confidence interval tells you how certain you are your facts are correct. The red areas on the graph are the margins of error.

construct a confidence interval

Construct a Confidence Interval : Steps

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.
Standard Deviation: 18.172.

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 how to construct a confidence interval using the t-distribution!

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18 thoughts on “Construct a Confidence Interval with the t-Distribution

  1. Angie Widdows

    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

    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:

    The standard deviation was obtained from the calculator in the link. I added the link to finding the mean…thanks for your suggestion,


  3. Jennifer Thomas

    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

    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

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

  6. Donna Allen

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

  7. April Fulton

    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

    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

    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

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

  11. Alison Bryant

    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

    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

    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.


  14. Andale

    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.


  15. Taylor Conley

    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!