Download - 7.Sampling and Sampling Distr
Chapter 7
Sampling and Sampling Distributions
Statistika
Descriptive statistics Collecting, presenting, and describing
data Inferential statistics
Drawing conclusions and/or making decisions concerning a population based only on sample data
Tools of Business Statistics
Population vs. Sample
a b c d
ef gh i jk l m n
o p q rs t u v w
x y z
Population Sample
b c
g i n
o r u
y
Less time consuming than a census Less costly to administer than a census It is possible to obtain statistical results of a
sufficiently high precision based on samples.
Why Sample?
Every object in the population has an equal chance of being selected
Objects are selected independently Samples can be obtained from a table of random
numbers or computer random number generators
A simple random sample is the ideal against which other sample methods are compared
Simple Random Samples
Making statements about a population by examining sample resultsSample statistics Population parameters (known) Inference (unknown, but can
be estimated from
sample evidence)
Inferential Statistics
Sample Population
Estimation e.g., Estimate the population mean
weight using the sample mean weight
Hypothesis Testing e.g., Use sample evidence to test
the claim that the population mean weight is 120 pounds
Inferential StatisticsDrawing conclusions and/or making decisions
concerning a population based on sample results.
A sampling distribution is a distribution of all of the possible values of a statistic for a given size sample selected from a population
Seperti distribusi sampling rata-rata
Sampling Distributions
Sampling Distributions ofSample Means
Sampling Distributions
Sampling Distribution of
Sample Mean
Sampling Distribution of
Sample Proportion
Sampling Distribution of
Sample Variance
Assume there is a population … Population size N=4 Random variable, X,
is age of individuals Values of X:
18, 20, 22, 24 (years)
Developing a Sampling Distribution
A B C D
Developing a Sampling Distribution
.25
0 18 20 22 24 A B C D
Uniform Distribution
P(x)
x
Summary Measures for the Population Distribution:
214
24222018
NX
μ i
2.236N
μ)(Xσ
2i
Now consider all possible samples of size n = 2
1st 2nd Observation Obs 18 20 22 24 18 18,18 18,20 18,22 18,24
20 20,18 20,20 20,22 20,24
22 22,18 22,20 22,22 22,24
24 24,18 24,20 24,22 24,24
16 possible samples (sampling with replacement)
1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
Developing a Sampling Distribution
16 Sample Means
Sampling Distribution of All Sample Means
1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21
20 19 20 21 22
22 20 21 22 23
24 21 22 23 24
18 19 20 21 22 23 240
.1
.2
.3 P(X)
X
Sample Means Distribution
16 Sample Means
_
Developing a Sampling Distribution
(no longer uniform)
_
Summary Measures of this Sampling Distribution:
Developing aSampling Distribution
iX 18 19+19 20 24E(X) 21 μ
N 16
2i
X
2 2 2
(X μ)σ
N
(18-21) (19-21) (24-21) 1.5816
Comparing the Population with its Sampling Distribution
18 19 20 21 22 23 240
.1
.2
.3 P(X)
X 18 20 22 24 A B C D
0
.1
.2
.3
PopulationN = 4
P(X)
X _
1.58σ 21μ XX 2.236σ 21μ
Sample Means Distributionn = 2
_
Let X1, X2, . . . Xn represent a random sample from a population
The sample mean value of these observations is defined as
Expected Value of Sample Mean
n
1iiX
n1X
Different samples of the same size from the same population will yield different sample means
A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean:
Note that the standard error of the mean decreases as the sample size increases
Standard Error of the Mean
nσσX
If a population is normal with mean μ and standard deviation σ, the sampling distribution of mean sampel (x-bar) is also normally distributed with
If the Population is Normal
μμX n
σσX
σx (μ, )n
Z-value for the sampling distribution of :
Z-value for Sampling Distributionof the Mean
where: = sample mean= population mean= population standard deviation
n = sample size
Xμσ
nσ
μ)X(σ
μ)X(ZX
We can apply the Central Limit Theorem: Even if the population is not normal, …sample means from the population will be
approximately normal as long as the sample size is large enough.
Properties of the sampling distribution:
and
If the Population is not Normal
μμx n
σσx
Central Limit Theorem
n↑As the sample size gets large enough…
the sampling distribution becomes almost normal regardless of shape of population
x
If the Population is not Normal
Population Distribution
Sampling Distribution (becomes normal as n increases)
Central Tendency
Variation
x
x
Larger sample size
Smaller sample size
Sampling distribution properties:
μμx
nσσx
xμ
μ
For most distributions, n > 25 will give a sampling distribution that is nearly normal
For normal population distributions, the sampling distribution of the mean is always normally distributed
How Large is Large Enough?
Dipunyai rata-rata nilai stat μ = 8 dan dev standard σ = 3. Diambil sampel random berukuran n = 36.
Berapakah probabilitas rata-rata sampel terletak antara 7.8 dan 8.2?
Sampel mean ~N(8, σ = 0.5)
Example
0.38300.5)ZP(-0.5
363
8-8.2
nσ
μ- μ
363
8-7.8P 8.2) μ P(7.8 XX
Dipunyai rata-rata berat timbang susu merk XYZ μ = 500,1 gr dan dev standard σ =1.3. Diambil sampel random berukuran n = 36.
Berapakah probabilitas rata-rata sampel kurang dari 500 gram ?
Sampel mean ~N(500,1, σ = 0.217)
Example
x -μ 500-500,1P(x 500) P σ 1,3n 36
P(Z 0.4615) 0.32