4.4 Normal Approximation to Binomial Distributions

Binomial Probability DistributionThe probability of k successes in n trials:

( ) ( ) (1 )k n knP X k p p

k

This represents getting exactly k successes.

E.g., Toss a coin 50 times. The probability of getting exactly 30 heads is

P(X = 30) = C(50, 30)(0.5)30(0.5)20

= 0.042

Let’s complicate things…• What is the probability of getting between

20 and 30 heads if you flip a coin 50 times?• This is a more complex situation:

= P(X = 20) + P(X = 21) + … + P(X=28)+ P(X = 29) + P(X = 30)

• Time-consuming calculation (even though numbers not very large)

• Use a graphical representation of binomial distribution

Go to the handout!

Copy Jarvis C.I./Pick Up/Data Management/Unit 4/XL_NormalApproximation.xls into your home drive and open it.

Follow directions on handout. We will discuss this in 15 minutes.

Normal Approximation of Binomial Distribution

• Binomial distributions can be approximated by normal distributions as long as the number of trials is relatively large.

• Binomial distributions– Display discrete random variables (whole

num.)

• Normal distributions– Display continuous values

Normal Approximation of Binomial Distribution

• To use normal distribution (continuous variables) to approx. binomial distributions (discrete variables):

• Use a range of values

• If X = 5, consider all values from 4.5 to 5.5

• If X = 3, 4, 5, consider all values from 2.5 to 5.5

• This is called a continuity correction

Why a range of values?

• Recall that probability is area under the curve

x

• P(X = x) = 0

• Create an area by extending x by 0.5 on either side

( 0.5, 0.5)x x

Recall

• How do we find the probability that a given range of values will occur?

• z-scores!x x

z

x np Why?

(1 )np p

( )x E X

Why? Just trust me.

Example 1What is the probability of getting between 20

and 30 heads if you toss a coin 50 times?(20 30)P X

x np50(0.5)25

(1 )np p 50(.5)(1 0.5)

3.5

x xz

x x

z

19.5 25

3.51.57

30.5 25

3.51.57

(19.5 30.5)P X

Example 1What is the probability of getting between 20

and 30 heads if you toss a coin 50 times?(20 30) (19.5 30.5)P X P X

( 30.5) ( 19.5)P X P X

( 1.57) ( 1.57)P z P z 0.9418 0.0582

0.8836

There is an 88% probability of getting between 20 and 30 heads – (binomial theorem gives 88.11%).

WARNING!• Not all binomial distributions can be

approximated with normal distribution

• Left- or right-skewed distributions don’t fit

• E.g. if n = 10, and p = 0.2

(3 4) ( 3) ( 4)P X P X P X

3 7 4 610 10(0.2) (0.8) (0.2) (0.8)

3 4

0.29

(Normal approx gives P(2.5 < X < 4.5) = 0.32 !)

Binomial Distribution

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

1 2 3 4 5 6 7 8 9 10

Number of successes

P(X

=k)

When can we approximate?

How can we tell whether the data is symmetrical enough to approximate using normal distribution?

If X is a binomial random variable of n independent trials, each with probability of success p, and if

np > 5 and n(1 – p) > 5

then X can be approximated by normal distribution.

Checking…• Example 1:

np = 50(0.5) n(1 – p) = 50(0.5) = 25 = 25 > 5 > 5Can be approximated by normal distribution

• Example 2:np = 10(0.2) n(1 – p) = 10(0.8) = 2 = 8 < 5 > 5Can’t be approximated by normal distribution