When can I use a normal distribution to approximate a binomial distribution?
- A binomial distribution can be approximated by a normal distribution provided
- n is large
- p is close to 0.5
- The mean and variance of a binomial distribution can be calculated by:
Why do we use approximations?
- These days calculators can calculate binomial probabilities so approximations are no longer necessary
- However it is easier to work with a normal distribution
- You can calculate the probability of a range of values quickly
- You can use the inverse normal distribution function (most calculators don't have an inverse binomial distribution function)
What are continuity corrections?
- The binomial distribution is discrete and the normal distribution is continuous
- A continuity correction takes this into account when using a normal approximation
- The probability being found will need to be changed from a discrete variable, X, to a continuous variable, XN
- For example, X = 4 for binomial can be thought of as for normal as every number within this interval rounds to 4
- Remember that for a normal distribution the probability of a single value is zero so
How do I apply continuity corrections?
- Think about what is largest/smallest integer that can be included in the inequality for the discrete distribution and then find its upper/lower bound
- You add 0.5 as you want to include k in the inequality
- You subtract 0.5 as you don't want to include k in the inequality
- You subtract 0.5 as you want to include k in the inequality
- You add 0.5 as you don't want to include k in the inequality
- For a closed inequality such as
- Think about each inequality separately and use above
- Combine to give
How do I approximate a probability?
- STEP 1: Find the mean and variance of the approximating distribution
- STEP 2: Apply continuity corrections to the inequality
- STEP 3: Find the probability of the new corrected inequality
- Use the "Normal Cumulative Distribution" function on your calculator
- The probability will not be exact as it is an approximate but provided n is large and p is close to 0.5 then it will be a close approximation
The random variable .
Use a suitable approximating distribution to approximate .
- In the exam, only use a normal approximation if the question tells you to. Otherwise use the binomial distribution.