1. 4.4 Normal Approximation to Binomial Distributions

2. Binomial Probability Distribution
The probability of k successes in n trials:
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

3. 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

4. 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.

5. 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

6. 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

7. Why a range of values? Recall that probability is area under the curve
P(X = x) = 0
Create an area by extending x by 0.5 on either side

8. Recall
How do we find the probability that a given range of values will occur?
z-scores!
Why?
Why?
Just trust me.

9. Example 1
What is the probability of getting between 20 and 30 heads if you toss a coin 50 times?

10. Example 1
What is the probability of getting between 20 and 30 heads if you toss a coin 50 times?
There is an 88% probability of getting between 20 and 30 heads – (binomial theorem gives 88.11%).

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

12. 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.

13. Checking… Example 1: Example 2: np = 50(0.5) n(1 – p) = 50(0.5)
= = 25
> > 5
Can be approximated by normal distribution
Example 2:
np = 10(0.2) n(1 – p) = 10(0.8)
= = 8
< > 5
Can’t be approximated by normal distribution