How to Find a Sample Size Given a Confidence Interval and Width

Confidence Intervals, sample size 9 Comments

This article shows you how to find a sample size for a given confidence interval and width (e.g. 95% interval, 6% wide) for an unknown population standard deviation.

Sample question: 41% of Jacksonville residents said that they had been in a hurricane. How many adults should be surveyed to estimate the true proportion of adults who have been in a hurricane, with a 95% confidence interval 6% wide?

Step 1: Using the data given in the question, figure out the following variables:

  • za/2: Divide the confidence interval by two, and look that area up in the z-table:
    .95 / 2 = 0.475
    The closest z-score for 0.475 is 1.96.
  • E (margin of error):  Divide the given width by 2.
    6% / 2
    = 0.06 / 2
    = 0.03
  • phat: use the given percentage.  41% = 0.41
  • qhat :  subtract phatfrom 1.
    1 – 0.41 = 0.59

Step 2:Multiply phatby qhat. Set this number aside for a moment.
0.41 × 0.59 =  0.2419

Step 3: Divide Za/2 by E.
1.96 / .03 = 65.3333333

Step 4: Square Step 3:
65.3333333 × 65.3333333 = 4268.44444

Step 5: Multiply Step 2 by Step 4:
0.2419 × 4268.44444 = 1,032.53671
= 1,033 people to survey.

Related posts:

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

  1. Angie Widdows said:

    Oct 08, 09 at 8:26 am

    This example was a tremendous help. The examples which give you line by line instructions really helps because it breaks everything down. I do not always necessarily understand why the steps are what they are but I do understand how to work them.

  2. Mary Johnson said:

    Oct 20, 09 at 5:52 pm

    This was helpful but I got a little confused on the order of operations. I have to do a “show me” on the homework and use the article.

  3. Catherine Flanagan said:

    Oct 21, 09 at 8:07 am

    This blog really helped me. I could not understand how to do some of the steps because the “show me” option on MathZone did not show every step. I like how the blogs breaks down every step.

  4. Sarah Joyner said:

    Oct 24, 09 at 3:29 pm

    It is very helpful how your site gives a break down of how to do the different problems. I learn best by example and this is a big help for me to understand stats!

  5. Donna Allen said:

    Oct 24, 09 at 5:22 pm

    I agree with Angie. Even though I may not understand all of the details as far as why so many steps are necessary, I do follow the explanations you give. Thank you!

  6. April Fulton said:

    Oct 25, 09 at 8:45 pm

    I am just really using this site because I have had trouble on the last two chapters and this site has helped me out a lot. I really like the step by step examples that the other site does not show us.

  7. Vanessa DuBarry said:

    Dec 14, 09 at 10:20 pm

    this was really helpful, i like how this blog is set up to show everything step by step and that is very helpful.

  8. Chaveli Jenkins said:

    Mar 03, 10 at 7:41 pm

    I was becoming very irritated, NOT anymore….Thanks to this forum….AWESOME….I looked on the internet, in my math book, on Mathzone and every thing broke down the problem but did not show HOW to get the Margin of Error (at least I didn’t see it) but this forum did AWESOME AWESOME AWESOME!!! I just got 5 encouragement points for the week after looking at this forum!!!!!

    ~Chaveli

  9. Alison Bryant said:

    Mar 22, 10 at 10:27 am

    This was extremely helpful, I really feel like this is one of the most clarifying aspects of the chapter. I was really having difficulty with the margin of error and this formula but I am now completely clear on the subject.

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