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Normal Approximation of a Binomial Distribution (Cambridge (CIE) A Level Maths): Revision Note

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Normal Approximation of Binomial

When can I use a normal distribution to approximate a binomial distribution?

  • A binomial distribution begin mathsize 16px style X tilde straight B left parenthesis n comma p right parenthesis end style can be approximated by a normal distribution X subscript N tilde straight N left parenthesis mu comma sigma squared right parenthesis  provided

    • n is large

    • p is close to 0.5

      • bold italic n bold italic p bold greater than bold 5

      • bold italic n bold italic q bold greater than bold 5 bold space bold italic w bold italic h bold italic e bold italic r bold italic e bold space bold italic q bold equals bold 1 bold minus bold italic p

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

    • mu equals n p

    • sigma squared equals n p left parenthesis 1 minus p right parenthesis

4-4-2-normal-approximation-of-binomial-diagram-1

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 3.5 less or equal than X subscript N less than 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 straight P left parenthesis 3.5 less or equal than X subscript N less than 4.5 right parenthesis equals straight P left parenthesis 3.5 less than X subscript N less than 4.5 right parenthesis

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 left parenthesis X equals k right parenthesis almost equal to P left parenthesis k space minus 0.5 less than X subscript N less than k plus 0.5 right parenthesis

  • P left parenthesis X less or equal than k right parenthesis almost equal to P left parenthesis X subscript N less than k plus 0.5 right parenthesis

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

  • P left parenthesis X less than k right parenthesis almost equal to P left parenthesis X subscript N less than k minus 0.5 right parenthesis

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

  • P left parenthesis X greater or equal than k right parenthesis almost equal to P left parenthesis X subscript N greater than k minus 0.5 right parenthesis

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

  • P left parenthesis X greater than k right parenthesis almost equal to P left parenthesis X subscript N greater than k plus 0.5 right parenthesis

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

  • For a closed inequality such as straight P left parenthesis a less than X less or equal than b right parenthesis

    • Think about each inequality separately and use above

    • P left parenthesis X greater than a right parenthesis almost equal to P left parenthesis X subscript N greater than a plus 0.5 right parenthesis

    • P left parenthesis X less or equal than b right parenthesis almost equal to P left parenthesis X subscript N less than b plus 0.5 right parenthesis

    • Combine to give

    • straight P left parenthesis a plus 0.5 less than X subscript N less than b plus 0.5 right parenthesis

How do I approximate a probability?

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

    • mu equals n p

    • begin mathsize 16px style sigma squared equals n p left parenthesis 1 minus p right parenthesis end style

  • 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:

      • n p greater than 5 

      • n p greater than 5 where q equals 1 minus p

Worked Example

The random variable X tilde B left parenthesis 1250 comma space 0.4 right parenthesis.

Use a suitable approximating distribution to approximate straight P left parenthesis 485 less or equal than space X space less or equal than space 530 right parenthesis.

3-4-2-normal-approximation-of-binomial-we-solution-with-addition

Examiner Tips and Tricks

  • 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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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.