How to Find a Confidence Interval (Population)

Confidence Intervals 14 Comments

Perhaps the trickiest part of confidence intervals is recognizing the various parts needed for the formula, like  z a/2. This article breaks everything down into simple steps and shows you how to find the most common confidence intervals for populations.
Sample question: 590 people applied to the Bachelor’s in Elementary Education program at Florida State College. Of those applicants, 57 were men. Find the 90% confidence interval of the true proportion of men who applied to the program.

Step 1: Read the question carefully and figure out the following variables:

  • α : subtract the given confidence interval from 1.
    1-.9=.10
  • z α/2: divide the given confidence interval by 2, then look up that area in the z-table.
    .9/2=.4500. The closest z-value to an area of .4500 is 1.65
  • phat: Divide the proportion given (i.e. the smaller number)by the sample size.
    57/510=0.112
  • qhat : Subtract phatfrom 1.
    1-0.112 = 0.888

Step 2: Multiply phatby qhat.
0.112 x 0.888 = 0.099456

Step 3: Divide step 2 by the sample size.
0.099456 / 510 = 0.000195011765

Step 4:Take the square root of step 3:
sqrt(0.000195011765) = 0.0139646613

Step 5: Multiply step 4 by z a/2:
0.0139646613 x 1.65 = 0.0230416911

Step 6:: For the lower percentage, subtract step 4 from phat.  0.112-0.0230416911 =  0.089 = 8.9%

Step 7:For the upper percentage, add step 4 to phat. 0.112 + 0.0230416911 =  0.135 = 13.5%


Related posts:

  1. How to Use a TI-83 to Find a Confidence Interval for the Population Proportion, p
  2. How to Find a Confidence Interval for a Mean (Unknown Population Standard Deviation)
  3. How to Find a Confidence Interval For a Sample
  4. How to Construct a Confidence Interval From Data Using the t-Distribution
  5. How to Find a Sample Size Given a Confidence Interval and Width
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14 Responses to “How to Find a Confidence Interval (Population)”

  1. Stacey Bell said:

    Oct 07, 09 at 4:08 pm

    This step by step was a tremendous help to me!

  2. Angie Widdows said:

    Oct 08, 09 at 8:24 am

    Wow. I didn’t realize there were so many steps involved. This example helped alot because it broke everything up, step by step. Was helpful to have the link to the Z-table.

  3. Lisa Barcomb said:

    Oct 18, 09 at 10:40 am

    Yeah these problems have a lot of steps to them but once you get them down they are really easy because you have the Z table. And in order for you to do your problems you have to know how to use the tables. Which in a way I thought they were fun.

  4. Mary Johnson said:

    Oct 20, 09 at 5:48 pm

    I can’t believe how much this helped me. I need the step by step explanation and this was great.

  5. Jennifer Foster said:

    Oct 22, 09 at 11:12 am

    z α/2: divide the given confidence interval by 2, then look up that area in the z-table.
    .9/2=.4500. The closest z-value to an area of .4500 is 1.65

    I don’t understand this….can someone please explain?
    What am I not getting?

  6. Stephanie said:

    Oct 22, 09 at 11:55 am

    Hi Jennifer,
    Why don’t you send me your working out, and I’ll tell you where you are going wrong (or at least, point out which specific part you don’t understand…is it how to look up a number in the z-table? Or something else?)
    Stephanie

  7. Jennifer Foster said:

    Oct 22, 09 at 6:23 pm

    The closest z-value to an area of .4500 is 1.65 …

    This is where I am really stuck. How did you get that? I can look up .45 on the table and it does not come out to be 1.65

  8. Stephanie said:

    Oct 23, 09 at 4:47 am

    Jennifer,
    What do you get when you look up .4500 in the center of the table? Let me know and I should be able to see where you are going wrong.
    Sometimes it helps to work backwards…look up 1.65 in the table (i.e. 1.6 in the left hand column and .05 in the top row). Do you see the .4505 where it intersects?
    Stephanie

  9. Donna Allen said:

    Oct 24, 09 at 5:26 pm

    Once again, I found this explanation very helpful. It was easier to follow your step by step instructions than the guided solution in mathzone.

  10. Jennifer Thomas said:

    Oct 25, 09 at 8:05 pm

    The step by step instructions are very helpful. Thank you!

  11. April Fulton said:

    Oct 25, 09 at 8:46 pm

    I have had the same question how do you get 1.65. But from this step by step and the teacher explaining to another student the same question, I see I am not the only one with these questions. This site is a big help!

  12. How to Do Everything Statistics » Finding Confidence Intervals for Two Populations (Proportions) said:

    Nov 01, 09 at 7:12 am

    [...] response): 67% or 0.67 Step 2: Find zα/2 (If you’ve forgotten how to find α/2, refer this article on confidence levels.) [...]

  13. Myrlande Blanc said:

    Aug 07, 11 at 4:06 am

    I was wondering how to get confidence interval with standard deviation 0.3, confidence level of 90%, and satisfaction score of 4.5

  14. Stephanie said:

    Sep 27, 11 at 4:54 pm

    Hi, Myrlande,
    Can you post your question on the forum? Unfortunately, I don’t have the time to answer math questions here.
    Thanks,
    Stephanie

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