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Wize University Statistics Textbook > Continuous Probability Distributions

Normal Approximation to Binomial

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Normal Approximation to Binomial

A random variable X∼B(n, p)X\sim B\left(n,\ p\right) that follows a binomial distribution is approximately normal (i.e. approximately follows a normal distribution) if the following conditions are met:
np>9   and  n(1−p)>9\boxed{np>9~~\text{ and}~~n(1-p)>9}
  • nn is the number of fixed trials
  • pp is the probability of success of any one individual trial

Note: this is another way of saying that nn has to be somewhat large.

Watch Out!
Some profs will use qq, which is 1−p1-p.
Some textbooks wil use the conditions np>10np>10 and n(1−p)>10n\left(1-p\right)>10 (or similar values).


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Standardization Formula

Wize Concept
Recall #1:
The standardization formula for a normal random variable XX is:
z=X−μσ\displaystyle \boxed{z=\frac{X-\mu}{\sigma}}

Recall #2:
Binomial mean=E(X)=μ=np     Binomial variance       =Var(X)=σ2=npq                       Binomial standard deviation        =SD(X)=σ=npq\begin{array}{}\text{Binomial mean}&&=E(X)=\mu=np\\\ \ \ \ \ \text{Binomial variance}&&\ \ \ \ \ \ \ =\text{Var}(X)=\sigma^2=npq\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{Binomial standard deviation}&&\ \ \ \ \ \ \ \ =\text{SD}(X)=\sigma=\sqrt{npq}\end{array}

If the conditions are met and we know that a binomial random variable can be approximated by a normal random variable, then our standardization formula becomes
z=x−npnpq\boxed{z=\frac{x-np}{\sqrt{npq}}}


About 70% of people have tried online dating.

a) If we randomly select 100 people, can we approximate a normal distribution?

b) What is the probability that at least 80 of the 100 have tried online dating?
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  1. Grab a piece of paper and try this problem yourself.
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Charles did not study for his exam, which has 40 multiple-choice questions. Each question has four answer choices – one being correct. He needs 50% to pass. If he randomly fills out the bubble sheet, what is the probability that he will pass the exam?


a) Normal Method:


b) Binomial method:

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Half of the population can swim. There is a 1:3 odd that more than p^\hat{p}% of my sample of 50 can swim. Find p^.\hat{p}.