Matakuliah
Tahun
: I0134 – Metode Statistika
: 2007
1

Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu :
• Mahasiswa akan dapat menghitung peluang Binomial dengan sebaran normal.
2

Outline Materi
• Metode deskriptif untuk sebaran normal
• pendekatan normal pada sebaran Binomial
• Pendekatan normal pada sebaran poisson
• Koreksi kekontinuan
3

The Normal Approximation to the Binomial
•
We can calculate binomial probabilities using
– The binomial formula
– The cumulative binomial tables
– Do It Yourself! applets
•
When n is large, and p is not too close to zero or one, areas under the normal curve with mean np and variance npq can be used to approximate binomial probabilities.
4

Approximating the Binomial

Make sure to include the entire rectangle for the values of x in the interval of interest. This is called the continuity correction.

Standardize the values of x using z
 x
 np npq

Make sure that np and nq are both greater than 5 to avoid inaccurate approximations!
5

Example
Suppose x is a binomial random variable with n = 30 and p = .4. Using the normal approximation to find P( x

10).
n = 30 p = .4 q = .6
np = 12 nq = 18
The normal approximation is ok!
Calculate
  np

30 (.
4 )

12
  npq

30 (.
4 )(.
6 )

2 .
683
6

Example
P ( x

10 )

P ( z

10 .
5

12
)
2 .
683

P ( z
 
.
56 )

.
2877
Applet
7

Example
A production line produces AA batteries with a reliability rate of 95%. A sample of n = 200 batteries is selected. Find the probability that at least 195 of the batteries work.
Success = working battery n = 200
The normal approximation is ok!
p = .95 np = 190 nq = 10
P ( x


195 )
P ( z


1 .
P ( z
46 )

194 .
5

190
200 (.
95 )(.
05 )
)

1

.
9278

.
0722
8

Sampling Distributions
Sampling distributions for statistics can be

Approximated with simulation techniques

Derived using mathematical theorems

The Central Limit Theorem is one such theorem.
Central Limit Theorem: If random samples of n observations are drawn from a nonnormal population with finite
 and standard deviation

, then, when n is large, the sampling distribution of the sample mean is approximately normally distributed, with mean
 and standard deviation

/ n x
. The approximation becomes more accurate as n becomes large.
9

Example
Toss a fair coin n = 1 time. The distribution of x the number on the upper face is flat or uniform.
   xp ( x )


1 (
1
6
)

2 (
1
6
)

...

6 (
1
6
)

3 .
5
 
( x
 
)
2 p ( x )

1 .
71
Applet
10

Example
Toss a fair coin n = 2 time. The distribution of x the average number on the two upper faces is moundshaped.
Mean :
 
3 .
5
Std Dev :

/ 2

1 .
71 / 2

1 .
21
Applet
11

Example
Toss a fair coin n = 3 time. The distribution of x the average number on the two upper faces is approximately normal.
Mean :
 
3 .
5
Std Dev :

/ 3

1 .
71 / 3

.
987
Applet
12

Why is this Important?
 The Central Limit Theorem also implies that the sum of n measurements is approximately normal with mean n
  n
 Many statistics that are used for statistical inference are sums or averages of sample measurements.
 When n is large, these statistics will have approximately normal distributions.
 This will allow us to describe their behavior and evaluate the reliability of our inferences.
13

How Large is Large?
If the sample is normal , then the sampling what the sample size.
When the sample population is approximately symmetric , the distribution becomes approximately normal for relatively small values of n. (ex. n=3 in dice example)
When the sample population is skewed , the sample size must be x before the sampling distribution of becomes approximately normal.
14

The Sampling Distribution of the Sample
Proportion

The Central Limit Theorem can be used to conclude that the binomial random variable x is approximately normal when n is large, with mean np and standard deviation . p
 x

The sample proportion, is simply a rescaling of the binomial random variable x , dividing it by n .

From the Central Limit Theorem, the sampling distribution of will also be approximately normal, with a rescaled mean and standard deviation.
15

The Sampling Distribution of the Sample
Proportion

A random sample of size n is selected from a binomial population with parameter p
.
 T he sampling distribution of the sample proportion, p
 x n pq
 will have mean p and standard deviation n

If n is large, and p is not too close to zero or one, the sampling distribution of will be approximately normal.
The standard deviation of p-hat is sometimes called the STANDARD ERROR (SE) of p-hat.
16

Finding Probabilities for the Sample Proportion

If the sampling distribution of is normal or approximately normal
, standardize or rescale the interval of interest in terms of z p

 pq p n

Find the appropriate area using Table 3.
Example: A random sample of size n = 100 from a binomial population with p = .4.
P (

.
5 )

P ( z

.
5

.
4
.
4 (.
6 )
)

P ( z

2 .
04 )
100

1

.
9793

.
0207
17

Example
The soda bottler in the previous example claims that only 5% of the soda cans are underfilled.
A quality control technician randomly samples 200 cans of soda. What is the probability that more than
10% of the cans are underfilled?
n = 200
S: underfilled can
p = P(S) = .05
P ( p

.
10 )

P ( z

.
10

.
05
.
05 (.
95 )
)

P ( z

3 .
24 )
q = .95
np = 10 nq = 190
OK to use the normal approximation

1

.
9994
200

.
0006
This would be very unusual, if indeed p = .05!
18

.
19