Estimasi Parameter
TIP-FTP-UB
Statistika Inferensial
Terbagi dua bagian:
 Estimasi (estimation)
 Uji hipotesis (test of hypotheses)
Estimasi
Terbagi menjadi dua bagian:
 Estimasi titik
 Estimasi interval
Point & Interval Estimation…
For example, suppose we want to estimate the mean
summer income of a class of business students. For
n=25 students, mean income is calculated to be 400
$/week.
point estimate
interval estimate
An alternative statement is:
The mean income is between 380 and 420 $/week.
10.4
Estimator
Qualities desirable in estimators:
 Unbiased
An unbiased estimator of a population parameter is an
estimator whose expected value is equal to that parameter.
Jadi jika 𝜃 adalah parameter dan 𝜃 adalah estimator unbiased
dari parameter 𝜃 apabila dipenuhi 𝐸 𝜃 = 𝜃.
 Consistent
An unbiased estimator is said to be consistent if the difference
between the estimator and the parameter grows smaller as the
sample size grows larger.
 Relatively efficient
If there are two unbiased estimators of a parameter, the one
whose variance is smaller is said to be relatively efficient.
Estimasi titik

Sebuah nilai tunggal yang digunakan untuk
mengestimasi sebuah parameter disebut
titik estimator (atau cukup estimator),
sedangkan proses untuk mengestimasi
titik disebut estimasi titik (point
estimation).
Estimasi Titik

Dapat dibuktikan bahwa 𝑋
adalah
estimator tak bias dari 𝜇 dan 𝑠 2 adalah
estimator tak bias dari 𝜎 2 .
Estimasi Interval
Proses
untuk
melakukan
estimasi
dengan
menggunakan interval disebut estimasi interval.
 Derajat kepercayaan dalam mengestimasi disebut
koefisien konfidensi.
 Misalnya 𝜃 merupakan estimator untuk parameter 𝜃,
sedangkan A dan B adalah nilai-nilai estimator
tersebut berdasarkan sampel tertentu, maka koefisien
kepercayaannya dinyatakan dengan:
𝑃 𝐴 <𝜃 <𝐵 =1−𝛼
diartikan bahwa kita merasa 100( 1 − 𝛼 )% percaya
(yakin) bahwa 𝜃 terletak diantara A dan B.
𝐴 < 𝜃 < 𝐵 disebut interval konfidensi (atau selang
konfidensi), sedangkan A dan B disebut batas-batas
kepercayaan. A disebut lower confidence limit, B disebut
upper confidence limit

Interval Konfidensi untuk Rataan 𝜇
Untuk 𝜎 2 diketahui
Jika 𝑋 adalah rataan sampel random berukuran n yang diambil dari
populasi normal (atau populasi tak normal dengan ukuran sampel
n≥ 30) dengan 𝜎 2 diketahui, maka interval konfidensi 100 1 −

Four commonly used confidence
levels…
Confidence Level

cut & keep handy!

Table 10.1
10.1
0
Example 1
A computer company samples demand during
lead time over 25 time periods:
235
421
394
261
386
374
361
439
374
316
309
514
348
302
296
499
462
344
466
332
253
369
330
535
334
It is known that the standard deviation of
demand over lead time is 75 computers. We
want to estimate the mean demand over lead
time with 95% confidence in order to set
inventory levels…
10.1
1
Example 1
“We want to estimate the mean demand
over lead time with 95% confidence in
order to set inventory levels”
IDENTIFY
Thus, the parameter to be estimated in the
population is mean.
And so our 𝜇 confidence interval estimator
will be:
10.1
2
Example 1
In order to use our confidence interval estimator, we need the
following pieces of data:
370.16
Calculated from the data…
1.96
75
n
Given
25
therefore:
The lower and upper confidence limits are 340.76 and 399.56.
10.1
3
Interval Width…
A wide interval provides little information.
For example, suppose we estimate with 95%
confidence that an accountant’s average starting
salary is between $15,000 and $100,000.
Contrast this with: a 95% confidence interval
estimate of starting salaries between $42,000 and
$45,000.
The second estimate is much narrower, providing
accounting students more precise information about
starting salaries.
10.1
4
Interval Width…
The width of the confidence interval
estimate is a function of the confidence
level, the population standard deviation, and
the sample size…
10.1
5
Interval Width
The width of the confidence interval estimate is a
function of the confidence level, the population
standard deviation, and the sample size
A larger confidence level
produces a w i d e r
confidence interval:
10.1
6
Interval Width…
The width of the confidence interval estimate
is a function of the confidence level, the
population standard deviation, and the sample
size…
Larger values of
produce w i d e r
confidence intervals
10.1
7
Interval Width
The width of the confidence interval estimate is a
function of the confidence level, the population standard
deviation, and the sample size
Increasing the sample size decreases the width of the
confidence interval while the confidence level can
remain unchanged.
Note: this also increases the cost of obtaining additional
data
10.1
8
Selecting the Sample Size
We can control the width of the interval by
determining the sample size necessary to produce
narrow intervals.
Suppose we want to estimate the mean demand “to
within 5 units”; i.e. we want to the interval estimate
to be:
Since:
It follows that
10.1
9
Selecting the Sample Size
Solving the equation
that is, to produce a 95% confidence interval
estimate of the mean (±5 units), we need to
sample 865 lead time periods (vs. the 25
data points we have currently).
10.2
0
Sample Size to Estimate a Mean
The general formula for the sample size
needed to estimate a population mean with
an interval estimate of:
Requires a sample size of at least this large:
10.2
1
Example 2
A lumber company must estimate the mean
diameter of trees to determine whether or
not there is sufficient lumber to harvest an
area of forest. They need to estimate this to
within 1 inch at a confidence level of 99%.
The tree diameters are normally distributed
with a standard deviation of 6 inches.
How many trees need to be sampled?
10.2
2
Example 2
Things we know:
Confidence level = 99%, therefore
We want
=.01
1 , hence W=1.
We are given that
= 6.
10.2
3
Example 2
We compute
That is, we will need to sample at least 239
trees to have a 99% confidence interval of
1
10.2
4