Edexcel A Level Maths: Statistics:復習筆記4.4.2 Normal Approximation of Binomial
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?
- 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
- For example,?X = 4?for binomial can be thought of as
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
Worked Example
The random variable 
.
Use a suitable approximating distribution to approximate 
.
Exam Tip
- In the exam, only use a normal approximation if the question tells you to. Otherwise use the binomial distribution.
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