Bài giảng Probability & Statistics - Lecture 7: Sampling - Bùi Dương Hải
7.2. Sampling Distribution A sampling distribution is a distribution of all of the possible values of a statistic for a given sample selected from a population Distribution of Sample Mean Distribution of Sample Proportion Distribution of Sample Variance
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Lecture 7. SAMPLING
Sampling
Sampling distribution
Point Estimate
Acceptance Interval
[1] Chapter 7. pp. 298 - 335
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Inferential Statistics
Inferential Statistics: Deduce information of
Population from Sample data.
Population size: , = ( , , , )
Parameters:
Population Mean
Population proportion
Proportion variance
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Example
Population:
Parameter: = 5.8 = 3.16
Proportion of “odd value” =
= 0.4
Sample 1:
̅ = 5.67,
= 4.33, ̅ =
= 0.33
Sample 2:
̅ = 7.25,
= 0.917, ̅ =
= 0
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4 8 5 7 6 7 6 2 5 8
4 8 5
8 6 7 8
7.1. Random Sample
Population data: from census
Exactly
Maybe impossible
Difficult to gather
Costly, much time
Sample data: from surveys
Possilbe to gather
Easier than census
Less cost and time
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Random Sample
Random Sample: Sample drawn from population that
every elements are selected with equal probability,
and independently.
For random sample
= ⋯ = =
= ⋯ = =
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Sampling Method
Stratified sampling
Cluster sampling
Systematic sampling
Convenience sampling
Judgment sampling
[1] p.331
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7.2. Sampling Distribution
A sampling distribution is a distribution of all of the
possible values of a statistic for a given sample
selected from a population
Distribution of Sample Mean
Distribution of Sample Proportion
Distribution of Sample Variance
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Distribution of Sample Mean
Sample Mean (Random Sample)
=̅
∑
i̅s random variable
̅ = ̅=
̅ = ̅
=
̅=
= Standard Error (S.E)
Sample mean has same expectation with , but
smaller variance.
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Correction Factor
In case of large sample or finite population, n is
relatively large in comparison to N
The correction factor is
̅ =
×
̅=
×
In lectures: sample is not large
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Normal Distribution
If Population is Normal distributed: ~ ( , ) or
Non-normal distributed but > 30 then:
~̅ ( ,̅ ̅
)
~̅ ,
Ex. Population ~ (20,4 )
Sample = 16
~̅ 20, 1
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Example
Example 7.1. Workers’ salary is Normal distributed with
Mean of 300 $ and Standard deviation of 20 $.
(a) What is the probability that salary of a worker chosen
randomly exceeds 305?
(b) Random choose 10 workers, what is the probability
that sample mean exceeds 305?
(c) What is the probability that sample mean of 100
workers exceeds 305?
(d) With the probability of 0.67, what is the maximum of
sample mean of 10 workers?
(e) With the probability of 0.67, what is the maximum of
sample mean of 100 workers?
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Distribution of Sample Proportion
Population Proportion = Probability =
Sample proportion = ̅
̅ = ; ̅ =
( )
With ≥ 100
~̅ ,
1 −
Example 7.2. Probability that candidate pass the exam is
0.4. Find the probability that proportion of pass in 200
candidate is greater than 45%
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7.3. Acceptance Interval
Assume that Population is known
Parameter , , are known
Deduce for statistics in sample
With probability of 95%, 90%, or (1 − )
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Sample mean
With probability of 95%
−1.96 < < 1.96 = 0.95
−1.96 <
−̅
/
< 1.96 = 0.95
Acceptance interval 95% of sample mean
− 1.96
< <̅ + 1.96
Or: ± 1.96
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Sample mean
In general, probability of 1 −
The acceptance interval for sample mean:
− /
< < + /
or ± /
Example 7.3. Worker income ($) is normal distribution
with mean of 300 and variance of 400. What is the
interval that average income of 25 workers falls into, with
probability of 95%, 90%, 80% ?
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Sample Proportion
If population parameter is known
is population proportion or probability
Acceptance interval of sample proportion ̅
− /
( − )
< < + /
( − )
Or
± /
( − )
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Sample Proportion
Example 7.4. Probability that a visitor buying at least one
item in the shopping mall is 0.3.
(a) Find the probability that in 200 visitors, there are at
least 65 customers.
(b) At probability level of 95%, in 200 visitors, what are
acceptance interval of relative frequency of number of
buyers, and acceptance interval of number of buyers?
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Key Concepts
Random Sample
Sampling Distribution
Acceptance Interval
[1] Chapter 7
(309) 13, 16,
(320) 21, 27, 29
(326) 37, 38
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Exercise
