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Pertemuan 10

Matakuliah : I0014 / BiostatistikaTahun : 2005Versi : V1 / R1

Pendugaan Parameter (II)

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Learning Outcomes

Pada akhir pertemuan ini, diharapkanmahasiswa akan mampu :

• Mahasiswa dapat menghitung pendugaan nilai tengah populasi (C3)

• Mahasiswa dapat menghitung pendugaan ragam populasi (C3)

• Mahasiswa dapat menghitung pendugaan proporsi populasi (C3)

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Outline Materi

• Pendugaan Nilai tengah

• Pendugaan Ragam

• Pendugaan Proporsi

1 2( dan )

2 2 21 2( dan / )

1 2( dan )p p p

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• Point Estimate – A single-valued estimate.– A single element chosen from a sampling distribution.– Conveys little information about the actual value of the

population parameter, about the accuracy of the estimate.

• Confidence Interval or Interval Estimate – An interval or range of values believed to include the

unknown population parameter.– Associated with the interval is a measure of the confidence we have that the interval does indeed contain the parameter of interest.

Jenis Penduga<<ISI>>

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We define as the z value that cuts off a right-tail area of under the standard normal curve. (1-) is called the confidence coefficient. is called the error probability, and (1-)100% is called the confidence level.

z2

2

zn

2

(1- )100% Confidence Interval:

x

Selang Kepercayaan (1-)100%

543210-1-2-3-4-5

0.4

0.3

0.2

0.1

0.0

Z

f(z)

Stand ard Norm al Distrib ution

z2

( )1

z2

2

2

<<ISI>>

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A (1-)100% confidence interval for when is not known (assuming a normally distributed population):

where is the value of the t distribution with n-1 degrees of

freedom that cuts off a tail area of to its right.

x t sn

2

t2

2

Selang Kepercayaan untuk bila Tidak Diketahui

<<ISI>>

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A large - sample (1 - )100% confidence interval for the population proportion,

where the sample proportion, p, is equal to the number of successes in the sample, ,divided by the number of trials (the sample size), , and q = 1 - p.

p:

p z

2 x

n

pqn

Penduga Selang untuk Proporsi

<<ISI>>

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A (1-)100% confidence interval for the population variance * (where the population is assumed normal):

where is the value of the chi-square distribution with n-1 degrees of freedom

that cuts off an area to its right and is the value of the distribution that

cuts off an area of to its left (equivalently, an area of to its right).

( ) , ( )n s n s

1 12

2

2

2

12

2

2

2

1

2

2

2

12

2

* Note: Because the chi-square distribution is skewed, the confidence interval for the population variance is not symmetric

Selang Kepercayaan untuk Ragam

<<ISI>>

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A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent random samples:

( )x x zs

n

s

n1 2

2

12

1

22

2

Selang Kepercayaan untuk Beda Dua Mean Populasi

<<ISI>>

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• When sample sizes are small (n1< 30 or n2< 30 or both), and both populations are normally distributed, the test statistic

• has approximately a t distribution with degrees of freedom given by (round downward to the nearest integer if necessary):

t x xsn

sn

( ) ( )1 2 1 2 0

12

1

22

2

df

sn

sn

sn

n

sn

n

12

1

22

2

2

12

1

2

1

22

2

2

21 1

<<ISI>>

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A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:df = (n1+ n2-2)

s n s n sn np

2 1 12

2 22

1 2

1 12

( ) ( )

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two samples. That is, larger weight is given to the variance from the larger sample.

Pendugaan Ragam Gabungan<<ISI>>

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A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and assuming equal population variances:

( )x x t sn np1 2

2

2 1

1

1

2

Selang Kepercayaan menggunakan Ragam Gabungan

<<ISI>>

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A (1-) 100% large-sample confidence interval for the difference between two population proportions:

( ) ( ) ( )

p p zp p

n

p p

n1 2

2

11

1

1

21

2

2

Selang Kepercayaan Beda Dua Proporsi

<<ISI>>

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A (1 - ) 100% confidence interval for 12

22

where F is the value obtained through the table and F1- is the left - tailed value of the distribution

obtained as the reciprocal of the F value with reversed - order degrees of freedom.

: ,

s

s

F

s

s

F

12

22

12

22

1

Selang kepercayaan Rasio Dua Ragam

<<ISI>>

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<< CLOSING>>

• Sampai saat ini Anda telah mempelajari pendugaan titik dan selang, baik untuk satu populasi maupun dua populasi

• Untuk dapat lebih memahami penggunaan pendugaan tersebut, cobalah Anda pelajari materi penunjang, dan mengerjakan latihan