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Pertemuan 16Pendugaan Parameter

Matakuliah : I0134 – Metoda Statistika

Tahun : 2005

Versi : Revisi

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Learning Outcomes

Pada akhir pertemuan ini, diharapkan mahasiswa

akan mampu :

• Mahasiswa dapat menghitung penduga selang dari rataan, proporsi dan varians.

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Outline Materi

• Selang nilai tengah (rataan)

• Selang beda nilai tengah (rataan)

• Selang proporsi dan beda proporsi

• Selang varians dan proporsi varians

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Interval Estimation

• Interval Estimation of a Population Mean:

Large-Sample Case• Interval Estimation of a Population Mean:

Small-Sample Case• Determining the Sample Size• Interval Estimation of a Population Proportion

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Interval Estimation of a Population Mean:Large-Sample Case

• Sampling Error

• Probability Statements about the Sampling Error

• Constructing an Interval Estimate:

Large-Sample Case with Known

• Calculating an Interval Estimate:

Large-Sample Case with Unknown

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Sampling Error

• The absolute value of the difference between an unbiased point estimate and the population parameter it estimates is called the sampling error.

• For the case of a sample mean estimating a population mean, the sampling error is

Sampling Error =| |x | |x

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Interval Estimate of a Population Mean:Large-Sample Case (n > 30)

• With Known

where: is the sample mean

1 - is the confidence coefficient

z/2 is the z value providing an area of

/2 in the upper tail of the standard

normal probability distribution

is the population standard deviation

n is the sample size

x zn

/2x zn

/2

xx

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Interval Estimate of a Population Mean:Large-Sample Case (n > 30)

• With Unknown

In most applications the value of the population standard deviation is unknown. We simply use the value of the sample standard deviation, s, as the point estimate of the population standard deviation.

x zsn

/2x zsn

/2

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Interval Estimation of a Population Mean:Small-Sample Case (n < 30) with Unknown

• Interval Estimate

where 1 - = the confidence coefficient

t/2 = the t value providing an area of /2 in the upper tail of a t distribution

with n - 1 degrees of freedom

s = the sample standard deviation

x tsn

/2x tsn

/2

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Contoh Soal: Apartment Rents

• Interval Estimation of a Population Mean:

Small-Sample Case (n < 30) with Unknown

A reporter for a student newspaper is writing an

article on the cost of off-campus housing. A sample of 10 one-bedroom units within a half-mile of campus resulted in a sample mean of $550 per month and a sample standard deviation of $60.

Let us provide a 95% confidence interval estimate of the mean rent per month for the population of one-bedroom units within a half-mile

of campus. We’ll assume this population to be normally distributed.

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• t Value

At 95% confidence, 1 - = .95, = .05, and /2 = .025.

t.025 is based on n - 1 = 10 - 1 = 9 degrees of freedom.

In the t distribution table we see that t.025 = 2.262.

Degrees Area in Upper Tail

of Freedom .10 .05 .025 .01 .005

. . . . . .

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

. . . . . .

Degrees Area in Upper Tail

of Freedom .10 .05 .025 .01 .005

. . . . . .

7 1.415 1.895 2.365 2.998 3.499

8 1.397 1.860 2.306 2.896 3.355

9 1.383 1.833 2.262 2.821 3.250

10 1.372 1.812 2.228 2.764 3.169

. . . . . .

Contoh Soal: Apartment Rents

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Estimation of the Difference Between the Means

of Two Populations: Independent Samples

• Point Estimator of the Difference between the Means of Two Populations

• Sampling Distribution

• Interval Estimate of Large-Sample Case

• Interval Estimate of Small-Sample Case

x x1 2x x1 2

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• Properties of the Sampling Distribution of – Expected Value

– Standard Deviation

where: 1 = standard deviation of population 1

2 = standard deviation of population 2

n1 = sample size from population 1

n2 = sample size from population 2

Sampling Distribution of x x1 2x x1 2

x x1 2x x1 2

E x x( )1 2 1 2 E x x( )1 2 1 2

x x n n1 2

12

1

22

2

x x n n1 2

12

1

22

2

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• Interval Estimate with 1 and 2 Known

where:

1 - is the confidence coefficient

• Interval Estimate with 1 and 2 Unknown

where:

Interval Estimate of 1 - 2:Large-Sample Case (n1 > 30 and n2 > 30)

x x z x x1 2 2 1 2 /x x z x x1 2 2 1 2 /

x x z sx x1 2 2 1 2 /x x z sx x1 2 2 1 2 /

ssn

snx x1 2

12

1

22

2 s

sn

snx x1 2

12

1

22

2

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• 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown

Substituting the sample standard deviations for the population standard deviation:

= 17 + 5.14 or 11.86 yards to 22.14 yards.

We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of 11.86 to 22.14 yards.

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

Contoh Soal: Par, Inc.

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Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)

• Interval Estimate with 2 Known

where:

x x z x x1 2 2 1 2 /x x z x x1 2 2 1 2 /

x x n n1 2

2

1 2

1 1 ( ) x x n n1 2

2

1 2

1 1 ( )

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• 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case

= 2.5 + 2.2 or .3 to 4.7 miles per gallon.

We are 95% confident that the difference between the

mean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car having the higher mpg).

sn s n s

n n2 1 1

22 2

2

1 2

2 21 12

11 2 56 7 1 8112 8 2

5 28

( ) ( ) ( . ) ( . ).s

n s n sn n

2 1 12

2 22

1 2

2 21 12

11 2 56 7 1 8112 8 2

5 28

( ) ( ) ( . ) ( . ).

x x t sn n1 2 025

2

1 2

1 12 5 2 101 5 28

112

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. ( ) . . . ( )x x t sn n1 2 025

2

1 2

1 12 5 2 101 5 28

112

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. ( ) . . . ( )

Contoh Soal: Specific Motors

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Inferences About the Difference Between the Proportions of Two

Populations

• Sampling Distribution of

• Interval Estimation of p1 - p2

• Hypothesis Tests about p1 - p2

p p1 2p p1 2

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• Expected Value

• Standard Deviation

• Distribution Form

If the sample sizes are large (n1p1, n1(1 - p1), n2p2,

and n2(1 - p2) are all greater than or equal to 5), thesampling distribution of can be approximatedby a normal probability distribution.

Sampling Distribution of p p1 2p p1 2

E p p p p( )1 2 1 2 E p p p p( )1 2 1 2

p pp pn

p pn1 2

1 1

1

2 2

2

1 1 ( ) ( ) p p

p pn

p pn1 2

1 1

1

2 2

2

1 1 ( ) ( )

p p1 2p p1 2

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Interval Estimation of 2

• Interval Estimate of a Population Variance

where the values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 - is the confidence coefficient.

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

( ) ( )

/ ( / )

n s n s

1 12

22

22

1 22

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• Chi-Square Distribution With Tail Areas of .025

95% of thepossible 2 values 95% of thepossible 2 values

22

00

.025.025.025.025

.9752.9752 .025

2.0252

Interval Estimation of 2

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• Selamat Belajar Semoga Sukses.