The normal approximation means that you can use the normal distribution as an approximation to calculate the probabilities for binomial distribution. Note that this can be done under the following conditions:
- The number of trials ‘n’ is large. (Generally values of n greater than 30 are considered to be large)
- The probability of success ‘p’ in a trial is sufficiently small.
- The mean ‘np’ is a finite number.
The below picture shows how the binomial distribution becomes closer and closer to the normal distribution as the number of trials increases.
Steps to apply the normal approximation to binomial distribution to calculate probabilities:
1. Given a random variable X, calculate the mean and variance using the formulae, Mean=np and Variance=np(1-p).
2. Check that the assumptions stated above for the validity of the approximation apply.
3. Treat the random variable as a normal variable with the mean and variance found in Step 1. But since we are using a continuous distribution (normal) to approximate a discrete (binomial) one, add a correction factor when calculating probabilities using the rules below.
4. The rules for the correction factor are:
P(X=n) gets replaced with P(n-0.5<X<n+0.5)
P(X<n) or P(X≤n) gets replaced with P(X<n+0.5)
P(X>n)or P(X ≥n) gets replaced with P(X>n-0.5)
5. Calculate the resulting probability by converting the normal distribution to standard normal distribution and using the Z table.
Examples:
1. Suppose a coin is thrown 100 times. Calculate the probability that you get heads 52 times using the normal approximation to the binomial distribution.
Solution: Let X denote the number of times we get heads in hundred tosses.
Then X follows binomial distribution with n=100 and p=1/2=0.5 and Mean= np =50.
Since the assumption for normal approximation are satisfied we may assume X follows normal distribution with mean=np=50 and variance=np(1-p)=25.
Then P(X=52) = P(51.5<x<52.5) (because of the correction factor)
=P (51.5-50/5< Z < 52.5-50/5) (converting normal to Z variate by subtracting mean and dividing by standard deviation)
=P(0.3<Z<0.5) = 0.073551 (by looking at the Z table)
2. Suppose a coin is thrown a 50 times. Calculate the probability that you get heads more than 36 times using the normal approximation to binomial distribution.
Solution: Let X denote the number of times we get heads in hundred tosses.
Then X follows binomial distribution with n=50 and p=1/2=0.5 and Mean= np =25.
Since the assumption for normal approximation are satisfied we may assume X follows normal distribution with mean=np=50 and variance=np(1-p)=12.5
Then P(X>36) = P(X>35.5) (because of the correction factor)
=P (Z > 35.5-25/√12.5) (converting normal to Z variate by subtracting mean and dividing by standard deviation)
=P(Z>2.9698) = 0.00149 (by looking at the Z table)
