How to Calculate the Minimum Binomial Sample Size

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A binomial question has two possible answers, like yes or no, black and white or positive and negative. For example, you may be interested in knowing if people support a local NFL team or not. To find the answer to this question, you don't have to survey the entire population; you just need to ask a sample of the population. In order to find out what the minimum sample size is that you need to produce an accurate result, you'll need to perform a few calculations.

Things You'll Need

  • Z-table (see Resources section)
  • Calculator
  • Identify the following from your question or data:

    zsub(a/2) (The z-value: divide the confidence interval by two and look up the value in the z-table.)
    E (the margin of error)
    Phat (percentage of respondents with positive responses)
    Qhat (1-Phat)

    For example, suppose the question is: "51 percent of residents say they support the local football team. How many residents should you survey with a 95 percent confidence interval 6 percent wide?"
    Your variables would be:

    zsub(a/2)=1.96 (.95/2=0.475. Look up 0.4750 in the z-table: see Resources section)
    E=0.03 (confidence interval of 0.06 divided by two)
    Phat=.51 (percentage of positive respondents expressed as a decimal)
    Qhat=.49 (1-Phat)

  • Multiply Phat by Qhat. In the above example, .51 x .49 = 0.2499. Set this number aside.

  • Divide zsub(a/2) by E. For the above example, zsub(a/2) divided by E = 1.96 / .03 = 65.3333.

  • Square Step 3. For our example, 65.333333 x 65.333333 = 4268.444444.

  • Multiply Step 2 by Step 4. 0.2499 x 4 268.444444 = 1,066.68427. You should survey 1,067 people.

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