Normal Approximation of a Binomial Distribution (Cambridge (CIE) AS 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 $X~\mathrm{B}(n,p)$ can be approximated by a normal distribution $X_N~\mathrm{N}(\mu ,{\sigma }^2)$  provided

  • n is large

  • p is close to 0.5

    • $\mathit{n}\mathit{p}\mathbf{>}\mathbf{5}$

    • $\mathit{n}\mathit{q}\mathbf{>}\mathbf{5}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\mathit{q}\mathbf{=}\mathbf{1}\mathbf{-}\mathit{p}$

• The mean and variance of a binomial distribution can be calculated by:

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

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, X_N

  • For example, X = 4 for binomial can be thought of as $3.5\le X_N<4.5$ 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 $\mathrm{P}(3.5\le X_N<4.5)=\mathrm{P}(3.5<X_N<4.5)$

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

• $P(X=k)\approx P(k-0.5<X_N<k+0.5)$

• $P(X\le k)\approx P(X_N<k+0.5)$

  • You add 0.5 as you want to include k in the inequality

• $P(X<k)\approx P(X_N<k-0.5)$

  • You subtract 0.5 as you don't want to include k in the inequality

• $P(X\ge k)\approx P(X_N>k-0.5)$

  • You subtract 0.5 as you want to include k in the inequality

• $P(X>k)\approx P(X_N>k+0.5)$

  • You add 0.5 as you don't want to include k  in the inequality

• For a closed inequality such as $\mathrm{P}(a<X\le b)$

  • Think about each inequality separately and use above

  • $P(X>a)\approx P(X_N>a+0.5)$

  • $P(X\le b)\approx P(X_N<b+0.5)$

  • Combine to give

  • $\mathrm{P}(a+0.5<X_N<b+0.5)$

How do I approximate a probability?

• STEP 1: Find the mean and variance of the approximating distribution

  • $\mu =np$

  • ${\sigma }^2=np(1-p)$

• 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:

    • $np>5$ 

    • $np>5$ where $q=1-p$