Point and Interval Estimates

Estimasi titik adalah nilai tunggal pada sampel (statistik) untuk mengestimasi nilai populasi (parameter).

Interval kepercayaan adalah kisaran nilai dimana terletak parameter populasi

Areas Under the Normal Curve

Between:± 1 - 68.26%± 2 - 95.44%± 3 - 99.74%

µµ-1σµ+1σ

µ-2σ µ+2σµ+3σµ-3σ

Jika kita mengambil observasi dari populasi terdistribusi normal, nilai yang terambil memiliki kemungkinan 68.26%) terletak dalam interval(µ-1σ, µ+1σ).

P((µ-1σ <x<µ+1σ) =0.6826.

P(µ-1σ <x<µ+1σ) vs P(x-1σ <µ <x+1σ)

P(µ-1σ <x<µ+1σ) adalah probabilitas bahwa x yang didapatkan dari sampel akan berada di antara interval (µ-1σ, µ+1σ).

P(x-1σ <µ <x+1σ) adalah probabilitas bahwa rata-rata populasi akan terletak di antara (x-1σ, x+1σ).

P(µ-1σ <x<µ+1σ) = P(µ-1σ -µ-x <x-µ-x<µ +1σ -µ-x) = P(-1σ -x <-µ<1σ -x)= P(-(-1σ -x )>-(-µ)>-(1σ -x))= P(1σ +x >µ>-1σ +x)= P(x-1σ <µ <x+1σ)

Elements of Confidence Interval Estimation

Confidence Interval

Sample Statistic

Confidence Limit (Lower)

Confidence Limit (Upper)

Probabilitas bahwa parameter populasi terletak di dalam interval.

Confidence Intervals

90% Samples

95% Samples

99% Samples

x_

nZX

XZX

X

XXXX 58.2645.1645.158.2

XX 96.196.1

Tingkat kepercayaan

1. Probabilitas bahwa parameter populasi yang tidak diketahui berada dalam interval

2. Dinotasikan (1 - tingkat kepercayaan adalah tingkat signifikansi:

probabilitas bahwa parameter tidak terdapat dalam interval

3. Nilai yang biasa dipakai 99%, 95%, 90%

Interpreting Confidence Intervals

Suatu interval kepercayaan bisa mengandung parameter populasi maupun tidak.

Untuk interval kepercayaan 95%, jika seluruh rata-rata sampel yang mungkin bisa didapatkan, maka 95% interval kepercayaan dari semua sampel itu akan mengandung rata-rata populasi.

Intervals & Level of Confidence

Sampling Distribution of Mean

Large Number of Intervals

Intervals Extend from

(1 - ) % of Intervals Contain .

% Do Not.

x =

1 - /2/2

X_

x_

XZX

XZX

to

Point Estimates and Interval Estimates

The factors that determine the width of a confidence interval are:

1. The size of the sample (n) from which the statistic is calculated.

2. The variability in the population, usually estimated by s.

3. The desired level of confidence.

nZX

XZX

)2/

()2/

(

Point and Interval Estimates

Jika standar deviasi populasi diketahui, atau

Jika sampel minimal 30 Gunakan distribusi z

n

szX

Point and Interval Estimates

Jika standar deviasi populasi tidak diketahui dan sampel kurang dari 30

Gunakan distribusi t

n

stX

Student’s t-Distribution

distribusi t adalah sekelompok distribusi yang simetrik dan berbentuk lonceng seperti distribusi normal standar tapi dengan area lebih besar di ekor. Tiap distribusit ditentukan oleh derajat kebebasannya (degrees of freedom – df). Semakin besar df, semakin distribusi t mendekati distribusi normal.

Zt

0

t (df = 5)

Standard Normal

t (df = 13)Bell-Shaped

Symmetric

‘Fatter’ Tails

Student’s t-Distribution

Upper Tail Area

df .25 .10 .05

1 1.000 3.078 6.314

2 0.817 1.886 2.920

3 0.765 1.638 2.353

t0

Student’s t Table

Assume:n = 3df = n - 1 = 2 = .10/2 =.05

2.920t Values

/ 2

.05

Derajat Kebebasan

Degrees of freedom (df) adalah jumlah nilai data independen tersedia untuk mengestimasi standar deviasi populasi. jika k parameter harus diestimasi sebelum standar deviasi populasi dapat dihitung dari sampel sebesar n, maka derajat kebebasan sama dengan n - k.

t-Values

where:= Sample mean= Population mean

s = Sample standard deviationn = Sample size

n

sx

t

x

Estimation Example Mean ( Unknown)

A random sample of n = 25 has = 50 and S = 8. Set up a 95% confidence interval estimate for .

X tS

nX t

S

nn n

/ , / ,

. .

. .

2 1 2 1

50 2 06398

2550 2 0639

8

2546 69 53 30

X

Standard Error of the Sample Means

Jika tidak diketahui dan n 30, standar deviasi sampel, s, digunakan untuk memperkirakan standar deviasi populasi:

n

ssx

95% and 99% Confidence Intervals for the sample mean

The 95% and 99% confidence intervals are constructed as follows:95% CI for the sample mean is given by

n

s96.1

n

s58.2

99% CI for the sample mean is given by

95% and 99% Confidence Intervals for µ

The 95% and 99% confidence intervals are constructed as follows:95% CI for the population mean is given by

n

sX 96.1

n

sX 58.2

99% CI for the population mean is given by

Constructing General Confidence Intervals for µ

In general, a confidence interval for the mean is computed by:

n

szX

EXAMPLE 3

The Dean of the Business School wants to estimate the mean number of hours worked per week by students. A sample of 49 students showed a mean of 24 hours with a standard deviation of 4 hours. What is the population mean?

The value of the population mean is not known. Our best estimate of this value is the sample mean of 24.0 hours. This value is called a point estimate.

Example 3 continued

Find the 95 percent confidence interval for the population mean.

12.100.2449

496.100.2496.1

n

sX

The confidence limits range from 22.88 to 25.12.About 95 percent of the similarly constructed intervals include the population parameter.

Confidence Interval for a Population Proportion

confidence interval untuk proporsi populasi diestimasi oleh:

n

ppzp

)1(

EXAMPLE 4

A sample of 500 executives who own their own home revealed 175 planned to sell their homes and retire to Arizona. Develop a 98% confidence interval for the proportion of executives that plan to sell and move to Arizona.

0497.35. 500

)65)(.35(.33.235.

Finite-Population Correction Factor

Untuk populasi yang diketahui jumlahnya (N), dengan sampel sebesar n, standard errors of the sample means perlu disesuaikan:

Standard error of the sample means:

1

N

nN

nx

Finite-Population Correction Factor

Standard error of the sample proportions:

1

)1(

N

nN

n

ppp

Penyesuaian ini disebut finite-population correction factor.

If n/N < .05, the finite-population correction factor dapat diabaikan.

EXAMPLE 5

Given the information in EXAMPLE 4, construct a 95% confidence interval for the mean number of hours worked per week by the students if there are only 500 students on campus.

Because n/N = 49/500 = .098 which is greater than 05, we use the finite population correction factor.

0648.100.24)1500

49500)(

49

4(96.124

Menentukan Besar Sampel

Terdapat 3 faktor yang menentukan besar sampel:Tingkat kepercayaan. Kesalahan maksimum yang diijinkan.Variasi populasi.

Menentukan Besar Sampel

Besar sampel (n):

where : E is the kesalahan yang dibolehkan, z adalah nilai z terkait tingkat kepercayaan yang dipilih, dan s adalah standar deviasi sampel pendahuluan.

2*

*

E

sznE

n

sz

nZX

XZX

)2/

()2/

(

EXAMPLE 6

A consumer group would like to estimate the mean monthly electricity charge for a single family house in July within $5 using a 99 percent level of confidence. Based on similar studies the standard deviation is estimated to be $20.00. How large a sample is required?

1075

)20)(58.2(2

n

Sample Size for Proportions

Menentukan besar sampel dalam kasus proporsi:

p adalah proporsi yang diestimasi berdasar pengalaman lalu atau survey pendahuluan; z adalah nilai z terkait tingkat kepercayaan yang dipilih; E adalah kesalahan maksimum yang ditoleransi.

2

)1(

E

Zppn

EXAMPLE 7

The American Kennel Club wanted to estimate the proportion of children that have a dog as a pet. If the club wanted the estimate to be within 3% of the population proportion, how many children would they need to contact? Assume a 95% level of confidence and that the club estimated that 30% of the children have a dog as a pet.

89703.

96.1)70)(.30(.

2

n

- END -

Chapter NineEstimation and Confidence IntervalsEstimation and Confidence Intervals