Download 4.4 Normal Approx to Binomial dist

4.4 Normal Approximation to
Binomial Distributions
Binomial Probability Distribution
The probability of k successes in n trials:
 n k
P( X  k )    ( p) (1  p) n k
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
• P(X = x) = 0
• Create an area
by extending x
by 0.5 on either
side
( x  0.5,xx  0.5)
Recall
• How do we find the probability that a given
range of values will occur?
• z-scores!
xx
z

x  np
  np(1  p)
Why?
Why?
x  E( X )
Just trust me.
Example 1
What is the probability of getting between 20
and 30 heads if you toss a coin 50 times?
P(20  X  30)  P(19.5  X  30.5)
x  np
 50(0.5)
 25
  np(1  p)
 50(.5)(1  0.5)
 3.5
z
xx

19.5  25

3.5
 1.57
z
xx

30.5  25

3.5
 1.57
Example 1
What is the probability of getting between 20
and 30 heads if you toss a coin 50 times?
P(20  X  30)  P(19.5  X  30.5)
 P( X  30.5)  P( X  19.5)
 P( z  1.57)  P( z  1.57)
 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
P(3  X  4)  P( X  3)  P( X  4)
10 
10 
3
7
   (0.2) (0.8)    (0.2) 4 (0.8) 6
3
4
Binomial Distribution
0.35
0.3
P(X=k)
0.25
0.2
0.15
0.1
 0.29
0.05
0
1
2
3
4
5
6
7
8
9
10
Number of successes
(Normal approx gives P(2.5 < X < 4.5) = 0.32 !)
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
>5
Can be approximated by normal distribution
• Example 2:
np = 10(0.2) n(1 – p) = 10(0.8)
=2
=8
<5
>5
Can’t be approximated by normal distribution