Using the Normal Approximation to solve a Binomial Problem

binomial distribution, Normal Distribution, Probability 10 Comments

When n * p and p * 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.

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 p * q = 228. 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

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 and your probability from step 5. 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-value.
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 96.49%.

Related posts:

  1. How to Solve a Binomial Distribution Problem Using the Binomial Formula
  2. How to solve a question about probability frequency distribution
  3. How to find a probability using a standard normal distribution
  4. Central Limit Theorem Problem Index
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10 Responses to “Using the Normal Approximation to solve a Binomial Problem”

  1. Vanessa DuBarry said:

    Oct 23, 09 at 4:32 am

    This problem helped me a lot, it really explained everything and I like how everything was explained step by step. The only problem I didnt understand and I wish it was explained here how did you get the p(x>290). I dont understand where the 290 came from.

  2. Stephanie said:

    Oct 23, 09 at 4:38 am

    Vanessa,
    The 290 came from the question: “find the probability that at least 290 are actually enrolled in school.”
    Stephanie

  3. Vanessa DuBarry said:

    Oct 23, 09 at 5:19 am

    oh my gosh nevermind I found it, I wasnt looking at the whole problem, I just get overwhelmed when I see all the numbers, but again this question really helped. it helped more then mathzone

  4. Vanessa DuBarry said:

    Oct 23, 09 at 5:20 am

    Hi professor I just saw your answer. thank you!

  5. Vanessa DuBarry said:

    Oct 23, 09 at 5:23 am

    Professor I have another question. you have that p times q is 228 but when I do it thats not the number I get, I get 0.2356

  6. Angie Widdows said:

    Oct 23, 09 at 9:39 am

    Wow. The amount of steps in this problem is amazing. I really liked this example because it broke out all of the steps. If I may add a suggestion to the site. It might be easier for the students if your examples went along with the week’s homework. It almost seems like we are working a week ahead on this site.

  7. Jennifer Thomas said:

    Oct 25, 09 at 8:31 pm

    Great example. This was helful. I sometimes get confused on which table I’m supposed to be using for each problem. I think I follow the order of each problem but don’t really understand it so I’m always unsure of which table we should be using.

  8. Lauren Schultz said:

    Nov 02, 09 at 7:21 am

    THIS HELPS ALOT. SOMETIMES I GOT CONFUSED ON WHICH TO USE FOR ‘P’ AND WHICH TO USE FOR ‘Q’ AND THAT THREW ME FOR A LOOP ON THE ANSWER.
    THANKS!!

  9. kate chorba said:

    Feb 23, 10 at 12:41 pm

    Prof,or anyone who knows, is there a way to do this problem using my ti-89 titanium calculator?

  10. jamie said:

    Apr 19, 11 at 6:01 pm

    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!

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