Document 15020148

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Matakuliah
Tahun
: L0104 / Statistika Psikologi
: 2008
Aplikasi Sebaran Normal
Pertemuan 12
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
Mahasiswa akan dapat menghitung peluang
Binomial dengan pendekatan pada sebaran
normal baku.
3
Bina Nusantara
Outline Materi
• Metode deskriptif untuk sebaran normal
• pendekatan sebaran Binomial pada sebaran
normal baku
• Koreksi kekontinuan
4
Bina Nusantara
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.
Bina Nusantara
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!
Bina Nusantara
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
np = 12
p = .4
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
Bina Nusantara
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)
Bina Nusantara
 P( z  1.46)  1  .9278  .0722
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.
Bina Nusantara
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
Bina Nusantara
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
Bina Nusantara
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.
Bina Nusantara
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
ˆ
The sample proportion, p  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.
Bina Nusantara
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.
Bina Nusantara
Finding Probabilities for
the Sample Proportion
If the sampling distribution of p̂ is normal or
approximately normal, standardize or rescale the
pˆ  p
interval of interest in terms of
z
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
Bina Nusantara
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!
Selamat Belajar
Semoga Sukses
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