Matakuliah
Tahun
Versi
: I0284 - Statistika
: 2005
: Revisi
Pertemuan 12
Aplikasi Sebaran Normal
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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
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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.
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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!
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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
np = 12
q = .6
nq = 18
The normal
approximation
is ok!
Calculate
  np  30(.4)  12
  npq  30(.4)(.6)  2.683
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Applet
Example
10.5  12
P( x  10)  P( z 
)
2.683
 P( z  .56)  .2877
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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
p = .95
np = 190
nq = 10
The normal
approximation
is ok!
194.5  190
P( x  195)  P( z 
)
200(.95)(.05)
 P( z  1.46)  1  .9278  .0722
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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 x is approximately
normally distributed, with mean  and standard deviation
 / n. The approximation becomes more accurate as n
becomes large.
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Example
Applet
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
1
 1( )  2( )  ...  6( )  3.5
6
6
6
  ( x   ) 2 p( x)  1.71
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Example
Applet
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
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Example
Applet
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
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Why is this Important?
The
Central Limit Theorem also implies that the
sum of n measurements is approximately normal with
mean n and standard deviation  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.
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How Large is Large?
If the sample is normal, then the sampling
distribution of x will also be normal, no matter
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 atxleast 30 before the sampling
distribution of becomes approximately normal.
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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 .
x
The sample proportion, pˆ  n is simply a rescaling
of the binomial random variable x, dividing it by n.
From the Central Limit Theorem, the sampling
distribution of p̂ will also be approximately normal,
with a rescaled mean and standard deviation.
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The Sampling Distribution of the
Sample Proportion
A random sample of size n is selected from a
binomial population with parameter p.
The sampling distribution of the sample proportion,
x
pˆ 
n
will have mean p and standard deviation
pq
n
If n is large, and p is not too close to zero or one, the
sampling distribution of p̂ will be approximately
normal.
The standard deviation of p-hat is sometimes called
the STANDARD ERROR (SE) of p-hat.
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Finding Probabilities for
the Sample Proportion
If the sampling distribution of p̂ is normal or
approximately normal, standardize or rescale the
interval of interest in terms of z  pˆ  p
pq
n
Find the appropriate area using Table 3.
.5  .4
Example: A random
P( pˆ  .5)  P( z 
)
.4(.6)
sample of size n = 100
100
from a binomial
population with p = .4.  P( z  2.04)  1  .9793  .0207
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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
q = .95
np = 10 nq = 190
OK to use the normal
approximation
P( pˆ  .10)
.10  .05
 P( z 
)  P ( z  3.24)
.05(.95)
200
 1  .9994  .0006
This would be very unusual,
if indeed p = .05!
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• Selamat Belajar Semoga Sukses.
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