Download Sampling, Sampling Distribution and Normality

4/17/2011
Tools of Business Statistics
Sampling, Sampling
Distribution and Normality
ƒ Descriptive statistics
ƒ Collecting, presenting, and describing data
ƒ Inferential statistics
Presented by:
Mahendra Adhi Nugroho, M.Sc
ƒ Drawing conclusions and/or making decisions
concerning a population based only on
sample data
Sources:
Anderson, Sweeney,wiliams, Statistics for Business and Economics , 6 e, Pearson education inc, 2007
Sugiyono, Statistika untuk penelitian, alfbeta, Bandung, 2007
Chap 7-2
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Populations and Samples
Population vs. Sample
ƒ A Population is the set of all items or individuals
of interest
ƒ
Examples:
All likely voters in the next election
parts p
produced today
y
All p
All sales receipts for November
a b
Examples:
Sample
cd
b
ef gh i jk l m n
x y
c
gi
o p q rs t u v w
ƒ A Sample is a subset of the population
ƒ
Population
o
z
n
r
u
y
1000 voters selected at random for interview
A few parts selected for destructive testing
Random receipts selected for audit
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-3
Why Sample?
Sampling Methods
Sampling
Methods
ƒ Less time consuming than a census
ƒ Less costly to administer than a census
Probability sampling
ƒ It is possible to obtain statistical results of a
sufficiently high precision based on samples.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-4
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
1. Simple random sampling
2. Proportionate stratified random
sampling
3. Disproportionate stratified
random sampling
4. Cluster sampling
Non probability sampling
1. Systematic sampling
2. Quota sampling
3. Incidental sampling
4. Purposive sampling
5. Surfeited sampling
6. Snowball sampling
Chap 7-5
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
1
4/17/2011
Inferential Statistics
Inferential Statistics
Drawing conclusions and/or making decisions concerning a
population based on sample results.
ƒ Making statements about a population by
examining sample results
ƒ Estimation
ƒ e.g., Estimate the population mean
weight using the sample mean
weight
Sample statistics
(known)
Population parameters
Inference
(unknown, but can
be estimated from
sample evidence)
Sample
ƒ Hypothesis Testing
ƒ e.g., Use sample evidence to test
the claim that the population mean
weight is 120 pounds
Population
Chap 7-7
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-8
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling Distributions of
Sample Means
Sampling Distributions
ƒ A sampling distribution is a distribution of
all of the possible values of a statistic for
a given size sample selected from a
population
Sampling
Distributions
Sampling
Distribution of
Sample
Mean
Sampling
Distribution of
Sample
Proportion
Sampling
Distribution of
Sample
Variance
Note: this chapter only discussing sample mean distribution.
Chap 7-9
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-10
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Developing a
Sampling Distribution
Developing a
Sampling Distribution
(continued)
Summary Measures for the Population Distribution:
ƒ Assume there is a population …
ƒ Population size N=4
A
B
C
∑X
D
μ=
variable X
X,
ƒ Random variable,
is age of individuals
=
ƒ Values of X:
18, 20, 22, 24 (years)
P(x)
i
N
18 + 20 + 22 + 24
= 21
4
σ=
∑ (X − μ)
i
N
.25
25
0
2
= 2.236
18
A
20
B
C
22
D
24
x
Uniform Distribution
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-11
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
Chap 7-12
2
4/17/2011
Developing a
Sampling Distribution
Developing a
Sampling Distribution
(continued)
(continued)
Now consider all possible samples of size n = 2
1st
Obs
18
2nd Observation
20
22
24
Sampling Distribution of All Sample Means
16 Sample
Means
Sample Means
Distribution
16 Sample Means
18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
1st 2nd Observation
Obs 18 20 22 24
1st 2nd Observation
Obs 18 20 22 24
22 22,18 22,20 22,22 22,24
18 18 19 20 21
18 18 19 20 21
24 24,18 24,20 24,22 24,24
20 19 20 21 22
20 19 20 21 22
22 20 21 22 23
22 20 21 22 23
16 possible samples
(sampling with
replacement)
24 21 22 23 24
Chap 7-13
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
_
P(X)
.3
.2
.1
0
24 21 22 23 24
18 19
20 21 22 23
(no longer uniform)
X
Chap 7-14
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Developing a
Sampling Distribution
_
24
Comparing the Population with its
Sampling Distribution
(continued)
Population
N=4
Summary Measures of this Sampling Distribution:
E(X) =
σX =
=
∑X
=
i
N
∑ ( X − μ)
i
μ = 21
18 + 19 + 21+ L + 24
= 21 = μ
16
6
Sample Means Distribution
n=2
μX = 21
σ = 2.236
σ X = 1.58
_
.3
P(X)
.3
.2
.2
.1
.1
P(X)
2
N
(18 - 21)2 + (19 - 21)2 + L + (24 - 21)2
= 1.58
16
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-15
18
A
20
B
C
22
D
24
X
0
18 19
20 21 22 23
_
24
X
Chap 7-16
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Standard Error of the Mean
Expected Value of Sample Mean
ƒ Let X1, X2, . . . Xn represent a random sample from a
population
ƒ The sample mean value of these observations is
defined as
X=
0
1 n
∑ Xi
n i=1
ƒ Different samples of the same size from the same
population will yield different sample means
ƒ A measure of the variability in the mean from sample to
sample is given by the Standard Error of the Mean:
σX =
σ
n
ƒ Note that the standard error of the mean decreases as
the sample size increases
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-17
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
Chap 7-18
3
4/17/2011
Z-value for Sampling Distribution
of the Mean
If the Population is Normal
ƒ Z-value for the sampling distribution of X :
ƒ If a population is normal with mean μ and
standard deviation σ, the sampling distribution
of X is also normally distributed with
μX = μ
σX =
and
Z=
σ
n
( X − μ) ( X − μ)
=
σ
σX
n
X = sample mean
where:
μ = population mean
σ = population standard deviation
n = sample size
Chap 7-19
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling Distribution Properties
Finite Population Correction
ƒ Apply the Finite Population Correction if:
ƒ a population member cannot be included more
than once in a sample (sampling is without
replacement), and
ƒ the
th sample
l iis llarge relative
l ti tto th
the population
l ti
(n is greater than about 5% of N)
ƒ Then
Var( X ) =
σ2 N − n
n N −1
or
σX =
σ
n
μx = μ
ƒ
(i.e.
x is unbiased )
N−n
N −1
Chap 7-21
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-20
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Sampling Distribution Properties
Normal Population
Distribution
μ
x
μx
x
Normal Sampling
Distribution
(has the same mean)
Chap 7-22
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
If the Population is not Normal
(continued)
ƒ We can apply the Central Limit Theorem:
ƒ For sampling with replacement:
As n increases,
ƒ Even if the population is not normal,
Larger
sample size
σ x decreases
ƒ …sample means from the population will be
approximately normal as long as the sample size is
large enough.
Smaller
sample size
Properties of the sampling distribution:
μ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x
Chap 7-23
μx = μ
and
σx =
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
σ
n
Chap 7-24
4
4/17/2011
Central Limit Theorem
If the Population is not Normal
(continued)
As the
sample
gets
size g
large
enough…
n↑
the sampling
distribution
becomes
almost normal
regardless of
shape of
population
Population Distribution
Sampling distribution
properties:
Central Tendency
μx = μ
Variation
σx =
σ
n
Larger
sample
size
Smaller
sample size
x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-25
How Large is Large Enough?
x
μx
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chap 7-26
Deciding Sample Size
Isaac and Michael Approach
ƒ For most distributions, n > 25 will give a
sampling distribution that is nearly normal
λ .N .P.Q
d (N − 1) + λ .P.Q
z Use table on sugiyono page 71
Nomogram Herry King
z Maximum sample size is 2000
2
s=
ƒ For normal population distributions
distributions, the
sampling distribution of the mean is always
normally distributed
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
x
μ
Sampling Distribution
(becomes normal as n increases)
2
2
Chap 7-27
Normality: Normal Curve
z Proper sample size in a research are 30-500
samples
z Proper size in categorized sample are minimum 30
samples each category
z Proper sample size for multivariate data analysis
(correlation or multivariate regression) are minimum
10 times of variables numbers (independent ad
dependent)
z Proper sample size for simple experimental design
that use experiment and control groups are 10 – 20
samples each variable group.
z A data set could have normal distribution if
sum of data and standard deviation of
upper mean and under mean data are
same.
same
29
Number
Suggested Sample Size
40
20
5
2
4
6
Value
8
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
10
12
5
4/17/2011
Normality: Normal Curve
Normality Test Using Chi square
z Divide normal curve in 6 areas base on
deviation standard values.
z Decide interval class. In this chase use 6
as class interval because chi square
normal distributions is divided in 6 part.
z Decide interval wide
z Counting the estimated chi square value
and compare the value with value that is
stated in chi square table.
2.7
%
13.53
%
2s
3s
34.53
%
1s
34.53
%
1s
13.53
%
2.7
%
2s
( − x)
z= x
i
s
3s
( f o − f e)
=
2
χ
2
f
e
Let me win!
If I cannot be a winner,
Let me brave in attempt!
© Mahendra Adhi Nugroho, M.Sc, Accounting Program Study of Yogyakarta State University
For internal use only!
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