Transcript ISU-ISU UTAMA PEMBANGUNAN DAERAH DI PROVINSI …

KULIAH STATISTIKA 2
CAKUPAN STATISTIKA
ESTIMATION METHODS
Estimation
Point
Estimation
Interval
Estimation
POINT ESTIMATION
Thinking Challenge
Suppose you’re
interested in the
average amount of
money that students
in this class (the
population) have on
them. How would
you find out?
Rp. 50.000
Or
Rp. 25.000 – Rp. 100.000
Term of estimation
–
–
–
–
Parameter
Statistic
Estimator
Estimate
Estimate Population
Parameter...
Mean

Proportion
p
Variance

Differences
with Sample
Statistic
x
p^
2
1 -  2
s
2
x1 -x2
What is point estimate?
A point estimate is a single
value (statistic) used to
estimate a population value
( p a r a m e t e r ) .
Ex:
ESTIMATION PROCESS
Population


Mean, , is
unknown




Sample





Random Sample
Mean


X
=
50

Ex:
The scores of 50 students of mid test value
Identify the target parameter and the point
estimator if 10 randomly choosen of student!
The scores of 50 students of mid test value Identify
the target parameter and the point estimator if 10
randomly chosen of student!
xˆ  x  E ( x)   
Randomly
chosen
55  71  76  77  85  ...  96
 81.5
10
(55  81,5) 2  (71 81,5) 2  ...  (96  81,5) 2
s  E (s )   
 146,056
9
2

2
2
39  48  63  .... 97  99
 79,98
50
How about population ?
(39  79,98) 2  (48  79,98) 2  ..... (97  79,98) 2  (99  79,98) 2
 
 152,387
49
2
CRITERIA FOR EVALUATING BEST
ESTIMATOR
A. Unbiased estimators
X or Xˆ and S 2
These random variables are
examples of statistics or
estimators

and
These fixed constants are
examples of parameter or
targets
X or Xˆ Is unbiased estimator of  If EX orE(Xˆ )  
E(S2)=2
2
True µ = E(X)
Bias = E(x) - µ
bias
µ
E(X)
B. Efficient estimators (minimum variance)
Estimators called efficient if the distribution of an
estimator to be highly concentrated or have a small
variance than another.
 2a
 2b
 2b<  2a
b efficient
estimator than a
Efficiency of u relatif to w  Var (w) / Var (u)
C. Consistent estimator
One of conditions that makes an estimator consistent is:
If its bias and variance both appraoach zero
Lim E(Xn) = µ
n
∞
and
Lim Var(Xn) = 0
n
∞
NOTE: Consisteny is more abstract, because it
defined as a limit: A consistent estimator is one
that concentrates in a narrower and narrower band
aroud its sample size n increases indefinitely .
Conclusion of Point Estimation
1. Provides a single value
• Based on observations from one
sample
2. Gives no information about how
close the value is to the unknown
population parameter
3. Example: Sample mean x = 3 is
point estimate of unknown
population mean
EXERCISE
1. Suppose each of the 200.000 adults in city under study has
eaten a number X of fast-food meals in the past week.
However, a residential phone survey on a week-day
afternoon misses those who are working-the very people
most likely to eat fast foods. As shown in the table below,
this leaves small population who would respond, especialy
small for higer values of X.
X= Number of
Meals
Whole target
(population)
Freq.
f
0
1
2
3
Total
Subpopulation
responding
Real. Freq.
f/N
Freq.
f
Real. Freq.
f/N
100.000
40.000
40.000
20.000
0,50
0,20
0,20
0,10
38,000
6,000
4,000
2,000
0,76
0,12
0,08
0,04
200.000
1,00
50.000
1,00
a. Find the mean µ of the whole targets population?
b. Find the sample mean  of the subpopulation who would
respond?
c. What is the estimator efficient of unbiased?
2. Suppose that a surveyor is traying to determine the area of a
rectangular field, in which the measured length X and the
measuered width Y are independent random variabeles that
fluctuate widely about the true values, according to the
following probability distribution
X
P(X)
Y
P(Y)
8
0.25
4
0.50
10
0.25
6
0.50
11
0.50
The calculte area A = XY of course is a random variable, and
is used to estimate the true area. If the true length and width
are 10 and 5, respectively,
a. Is X an unbiased estimator of the true length?
b. Is Y an unbiased estimator of the true width?
c. Is A an unbiased estimator of the true area?