Statistics How To

Normal Approximation to solve a Binomial Problem

Binomial Theorem > Normal Approximation to solve a binomial problem

Normal Approximation: Overview

normal approximation

A normal distribution curve, sometimes called a bell curve.


When n * p and n * q are greater than 5, you can use the normal approximation to solve a binomial distribution problem. This article shows you how to solve those types of problem using the continuity correction factor.

Normal Approximation: Example #1 (Video)

Normal Approximation: Example#2

Sixty two percent of 12th graders attend school in a particular urban school district. If a sample of 500 12th grade children are selected, find the probability that at least 290 are actually enrolled in school.
Step 1: Determine p,q, and n:
p is defined in the question as 62%, or 0.62
To find q, subtract p from 1: 1 – 0.62 = 0.38
n is defined in the question as 500

Step 2: Determine if you can use the normal distribution:
n * p = 310 and n * q = 190. These are both larger than 5.

Step 3: Find the mean, μ by multiplying n and p:
n * p = 310

Step 4: Multiply step 3 by q :
310 * 0.38 = 117.8.

Step 5: Take the square root of step 4 to get the standard deviation, σ:
sqrt(117.8)=10.85
Note: The formula for the standard deviation for a binomial is &sqrt;(n*p*q).

Step 6: Write the problem using correct notation:
P(X≥290)

Step 7: Rewrite the problem using the continuity correction factor:
P (X ≥ 290-0.5)= P (X ≥ 289.5)

Step 8: Draw a diagram with the mean in the center. Shade the area that corresponds to the probability you are looking for. We’re looking for X ≥ 289.5, so…
graph using continuity correction factor

Step 9: Find the z-score.
You can find this by subtracting the mean (μ) from the probability you found in step 7, then dividing by the standard deviation (σ): (289.5 – 310) / 10.85 = -1.89
Step 10: Look up the z-value in the z-table:
The area for -1.819 is 0.4706.
Step 11: Add .5 to your answer in step 10 to find the total area pictured:
0.4706+ 0.5 = 0.9706.

That’s it! The probability is .9706, or 97.06%.

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Normal Approximation to solve a Binomial Problem was last modified: October 25th, 2017 by Stephanie

24 thoughts on “Normal Approximation to solve a Binomial Problem

  1. jamie

    oh my goodness, i feel like kissing you right now! thankyou thankyou thankyou! i will be buying the book for my end semester exam!!!
    God bless you!

  2. rc

    Professor, there is something I don’t get in this example – how can p*q be >5? since they are both probabilities, this means they are maximum 1 each (and not in the same time :), so p*q can’t be >1 ever (in fact, I guess the max p*q =.25)

  3. Zeshan

    If n*p is less than 30 or in this case <5, then instead of normal distribution, you would be using t-distribution as the central limit theorem does not apply. t-distribution is similar but you would use the t-table chart and find the t-value instead of the z-value and work it the same way after

  4. hunain turab

    prof plz tell me the procedure to solve this problem by using normal distribution
    if p(x>a)=0.4995 then find the value of a

  5. Eddie

    Wouldn’t it just be easier to find the z-score of -1.89
    And then subtract it. So it would be 1-.0294

    The last two steps confuse me greatly.

  6. Andale

    Eddie,
    There’s more than one way to get the answer in stats. This is the way I taught my students, but do whatever works for you :)
    Stephanie

  7. Meliza

    Well I do have a question how would we find the t-table and will the answer be correct since my z table stops at 3.49

  8. Prasant Kumar

    z- score of -1.819 is 0.0294 and why did you add 0.5 in that. we can simply subtract this z-value from 1 and we will get the answer as z-score is always calculate CDF (area to the left of the curve). Since we are interested in knowing area to the right of the curve.

  9. Andale

    The z-score isn’t always the whole area to the left of the curve. Sometimes it’s the area between the mean and the score (as in this case). You have to add .5 to account for the area to the left of the mean.

  10. bilal

    hi prof.plz guide me when should we use correction factor and when not we need to use this?how can v differentiate?…….

  11. Andale

    You use it when you have a discrete function and you want to use a continuous function (usually a binomial to a normal or, more rarely, a Poisson with a large λ to a normal). These are the only two cases I can think of where you would want to use it…

  12. Lim Souchhy

    Excuse me, I want to ask you something.
    Actually i did an exercise about this topic.
    And I got the answer
    a. P=77.29% by binomial
    b. P=76.94% by normal approximation
    The question is how do I comment on answer (b) to exact percentage found in answer (a)?

  13. Andale

    Lim,
    I’m not sure what you mean by “The question is how do I comment on answer (b) to exact percentage found in answer (a)?”. Could you rephrase your question?
    Thanks.

  14. Lim Souchhy

    I mean “how to comment on (b) to exact percentage found in answer (a)”?
    This exercise requires to calculate the percentage of probability by two ways. Question (a) calculate by binomial, Question (b) calculate by normal approximation plus compare and explain and comment why it is similar or equal to (a). But I can only calculate and cannot comment on it.

  15. Andale

    OK, if I have this right, you have found (a), your binomial, and (b) the normal approximation. What are your results? I’m assuming they are different: the question is probably asking you to say what the difference is and why (i.e. because you’re using the two different methods).

  16. Salmi Tchitumba

    OMG….. at first it was kind of difficult…… But i got it now.. thanks a lot

  17. Eric Ndofor

    Dear Professor,
    Both examples are very clear up to the point of z tables.
    I noticed you used the A-3 Table for both examples on this page. What is not clear to me is what criteria you use to determine what column on the Z table can be used. It is not clear to me why you used Column D for question 1 and column A for example 2. Thanks for explaining why.