Chapter 6: The Normal Distribution and The Central Limit Theorem
Chapter 6: The Normal Distribution and The Central Limit Theorem
6.5 The Normal Approximation to the Binomial
Learning Objectives
By the end of this section, the student should be able to:
- Approximate the binomial distribution using the normal distribution
The binomial formula is cumbersome when the sample size (n) is large, particularly when we consider a range of observations. Consider the following example.
Example
Approximately 15% of the US population smokes cigarettes. A local government believed their community had a lower smoker rate and commissioned a survey of 400 randomly selected individuals. The survey found that only 42 of the 400 participants smoke cigarettes. If the true proportion of smokers in the community was really 15%, what is the probability of observing 42 or fewer smokers in a sample of 400 people?
Solution
We first need to verify the four conditions for the binomial model are met. The question posed is equivalent to asking, what is the probability of observing k = 0, 1, 2, …, or 42 smokers in a sample of n = 400 when p = 0.15? We can compute these 43 different probabilities and add them together.
[latex]P(k = 0 \text{ or } k = 1 \text{ or } ... \text{ or } k = 42)[/latex]
[latex]= P(k = 0) + P(k = 1) + ... + P(k = 42)[/latex]
[latex]= 0.0054[/latex]
The computations in the previous example are tedious, long, and near impossible if you do not have access to technology. Luckily, we have discovered the [latex]\text{binomcdf()}[/latex] function previously, so this problem with the technology is not horrible to do (Try it for yourself: [latex]\text{binomcdf}(400, 0.15, 42) = 0.0054[/latex]).
In some cases we may use the normal distribution as an easier and faster way to estimate binomial probabilities. In general, we should avoid such work if an alternative method exists that is faster, easier, and still accurate. Recall that calculating probabilities of a range of values is much easier in the normal model. We might wonder, is it reasonable to use the normal model in place of the binomial distribution? Surprisingly, yes, if certain conditions are met.
Historical Note
Historically, being able to compute binomial probabilities was one of the most important applications of the central limit theorem. Binomial probabilities with a small value for n (say, 20) were displayed in a table in a book. To calculate the probabilities with large values of n, you had to use the binomial formula, which could be very complicated. Using the normal approximation to the binomial distribution simplified the process.
To compute the normal approximation to the binomial distribution, take a simple random sample from a population. You must meet the conditions for a binomial distribution:
- there are a certain number n of independent trials
- the outcomes of any trial are success or failure
- each trial has the same probability of a success p
Recall that if X is the binomial random variable, then [latex]X \sim B(n, p)[/latex]. The shape of the binomi