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Construct a Confidence Interval with the T-Distribution or Z-Distribution

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Contents:
1. Confidence Interval (T-Distribution)
2. Confidence Interval (Z-Distribution)


Construct a Confidence Interval with a t-distribution

Watch the video or read the steps below:

You may be asked in elementary statistics to construct a confidence interval from a given set of data. If you have one small set of data (under 30 items), you’ll want to use the t-distribution instead of the z-distribution to construct your confidence interval. Thanks to the t-distribution table, only a few short steps are needed to calculate the t-distribution.

The formula for constructing a CI with the t-distribution.

The formula for constructing a CI with the t-distribution.

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, μ and standard deviation, σ for the data.
σ: 18.172.
μ: 71
Set these numbers aside for a moment.

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. Set this number aside for a moment.

Step 3: Subtract the confidence level from 1, then divide by two. This is your alpha level.
(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 5: Divide your std dev (step 1) by the square root of your sample size.
18.172 / √(10) = 5.75

Step 6: : Multiply step 4 by step 5.
2.821 × 5.75 = 16.22075

Step 7: For the lower end of the range, subtract step 6 from the mean (Step 1).

71 – 16.22075 = 54.77925

Step 8: For the upper end of the range, add step 6 to the  mean (Step 1).
71 + 16.22075 = 87.22075

That’s how to construct a confidence interval using the t-distribution!


Confidence Interval with the Z-Distribution

Watch the video or read the article below:

If you don’t know your population mean (μ) but you do know the standard deviation (σ), you can find a confidence interval for the population mean, with the formula:
x̄ ± z* σ / (√n),

Sample problem: Construct a 95 % confidence interval an experiment that found the sample mean temperature for a certain city in in August was 101.82, with a population standard deviation of 1.2. There were 6 samples in this experiment.

Step 1: Subtract the confidence level (Given as 95 percent in the question) from 1 and then divide the result by two. This is your alpha level, which represents the area in one tail.
(1 – .95) / 2 = .025

Step 2: Subtract your result from Step 1 from 1 and then look that area up in the middle of the z-table to get the z-score:

  1. 1 – 0.25 = 0.975
  2. z score = 1.96.

Step 3: Plug the numbers into the second part of the formula and solve:
z* σ / (√n)
= 1.96 * 1.2/√(6)
= 1.96 * 0.49
= 0.96

Step 4: For the lower end of the range, subtract step 3 from the mean.
101.82 – 0.96 = 100.86

Step 5: For the upper end of the range, add step 3 to the  mean.
101.82 + 0.96 = 102.78.

The CI is (100.86,102.78)

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18 thoughts on “Construct a Confidence Interval with the T-Distribution or Z-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:
    http://www.jerrydallal.com/LHSP/dof.htm

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

    Stephanie

  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.

    Stephanie

  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.

    Stephanie

  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!