# 3.4.2 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?

• 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
• 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
• 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

#### Worked example

The random variable

Use a suitable approximating distribution to approximate .

#### Exam Tip

• 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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