Download Normal approximation of Binomial Distribution

Normal Approximation to
Binomial Distribution
Consider the binomial distribution
with n trials, and probability of
success is p
This distribution is approximately
normal if
• np > 5 and nq > 5.
In this case it is approximated by a
normal distribution with
• Mean = np and Variance = npq
Binomial Histogram (n=25, p=.95)
0.4
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
24
22
20
18
16
14
12
10
8
6
4
2
0
Number of Successes
This binomial distribution doesn’t look approximately
normal (it is not bell shaped).
Note np = 23.75 > 5 but nq = 1.25 < 5.
Binomial (n=25, p=.7) and Normal Approximation
0.18
0.16
Probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
3
5
7
9
11
13
15
17
19
21
23
25
Number of Success
The normal distribution is a good approximation:
np = (25)(.7) = 17.5 > 5, nq = (25)(.3) = 7.5 > 5
Binomial (n=40, p=.8) and Normal Approximation
0.18
0.16
Probability
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
4
7
10
13
16
19
22
25
28
31
34
Number of Success
The normal is a good approximation:
np = 32> 5 and nq = 8 > 5.
37
40
Binomial (n=100, p=.4) and Normal Approximation
0.09
0.08
Probability
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1
5
9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69
Number of Success
The normal is a good approximation:
np = 40> 5 and nq = 60 > 5.
Ex: For the following distribution, estimate P(9 < r < 13) by
(i) using the binomial probability formula directly;
(ii) using the normal approximation.
Continuity Correction
Overbooked Flights
• Have you ever
arrived at the
airport and
found that your
flight was over
booked?