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Normal Approximation of Binomial (CIE A Level Maths: Probability & Statistics 1)
Revision Note
Normal Approximation of Binomial
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?
- If there are a large number of values for a binomial distribution there could be a lot of calculations involved and it is inefficient to work with the binomial distribution
- 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)
- In your exam you must use the formula and not a calculator to find binomial probabilities so you are limited to small values of n
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
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- 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
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- You subtract 0.5 as you want to include k in the inequality
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- 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
- Find the standard normal probability and use the table of the normal distribution
- 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
- To decide if n is large enough and if p is close enough to 0.5 check that:
- where
- To decide if n is large enough and if p is close enough to 0.5 check that:
Worked example
The random variable
Use a suitable approximating distribution to approximate .
Examiner Tip
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In the exam, the question will often tell you to use a normal approximation but sometimes you will have to recognise that you should do so for yourself. Look for the conditions mentioned in this revision note, n is large, p is close to 0.5, np > 5 and nq > 5.
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