Lecture 8 - ENEE1006IU

lOMoARcPSD|364 906 32 APPLIED STATISTICS COURSE CODE: ENEE1006IU Lecture 8:
Chapter 4: Probability and Distribution
(3 credits: 2 is for lecture, 1 is for lab-work)
Instructor: TRAN THANH TU Email: tttu@hcmiu.edu.vn tttu@hcmiu.edu.vn 1 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•The sampled population is the population from which the sample is drawn.
•The target population is the population we want to make inferences about
•A frame is a list of the elements that the sample will be selected from. The sampling problems Selecting a sample Point estimation
Introduction to sampling distributions Sampling distribution of Sampling distribution of
Properties of point estimators tttu@hcmiu.edu.vn 2 lOMoARcPSD|364 906 32 Other sampling methods
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
Numerical characteristics of a population are called parameters. Sampling problems: -Sampling errors
-Lack of sample representativeness
-Difficulty in estimation of sample size
-Lack of knowledge about the sampling process -Lack of resources -Lack of cooperation
-Lack of existing appropriate sampling frames for larger population tttu@hcmiu.edu.vn 3 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Selecting a sample:
Sampling from a Finite Population: A simple random sample of size n from a
finite population of size N is a sample selected such that each possible sample of
size n has the same probability of being selected.
One procedure for selecting a simple random sample from a finite population is
to use a table of random numbers to choose the elements for the sample one at
a time in such a way that, at each step, each of the elements remaining in the
population has the same probability of being selected.
Sampling n elements in this way will satisfy the definition of a simple random
sample from a finite population. tttu@hcmiu.edu.vn 4 lOMoARcPSD|364 906 32 sampling without replacement sampling with replacement
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
we cannot develop a list of all the elements that could •Selecting a sample:
be produced, the population is considered infinite
Sampling from an infinite Population: A random sample of size n from an
infinite population is a sample selected such that the following conditions are satisfied:
1. Each element selected comes from the same population.
2. Each element is selected independently. tttu@hcmiu.edu.vn 5 lOMoARcPSD|364 906 32
Care and judgment must be exercised in implementing the selection process for
obtaining a random sample from an infinite population. prevent selection bias tttu@hcmiu.edu.vn 6 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Point estimation:
To estimate the value of a population parameter, we compute a corresponding
characteristic of the sample, referred to as a sample statistic.
•By making the preceding computations, we perform the statistical procedure called point estimation. •We refer to:
- the sample mean as the point estimator of the population mean µ
- the sample standard deviation s as the point estimator of the population standard deviation , and 7 lOMoARcPSD|364 906 32
- the sample proportion as the point estimator of the population proportion p. The
numerical value obtained for , s, or is called the point estimate. tttu@hcmiu.edu.vn
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Introduction to sampling distributions:
•If we consider the process of selecting a simple random sample as an experiment,
the sample mean is the numerical description of the outcome of the experiment.
the sample mean is a random variable
has a mean or expected value, a standard deviation, and a probability distribution. tttu@hcmiu.edu.vn 8 lOMoARcPSD|364 906 32
•Because the various possible values of are the result of different simple random
samples, the probability distribution of is called the sampling distribution of .
•Knowledge of this sampling distribution and its properties will enable us to make
probability statements about how close the sample mean is to the population mean µ.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of : the probability distribution of all possible values of the sample mean .
The sampling distribution of has an expected value or mean, a standard tttu@hcmiu.edu.vn 9 lOMoARcPSD|364 906 32
Infinite (population is large, some how can’t be identified) tttu@hcmiu.edu.vn 10 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of : the probability distribution of all possible values of the sample mean .
Form of the Sampling Distribution of :
-Population has a normal distribution: the sampling distribution of is normally
distributed for any sample size.
- Population does not have a normal distribution: central limit theorem is
helpful in identifying the shape of the sampling distribution of tttu@hcmiu.edu.vn 11 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of : the probability distribution of all possible values of the sample mean .
Practical Value of the Sampling Distribution of tttu@hcmiu.edu.vn 12 lOMoARcPSD|364 906 32
Whenever a simple random sample is selected and the value of the sample
mean is used to estimate the value of the population mean µ, we cannot
expect the sample mean to exactly equal the population mean.
The practical reason we are interested in the sampling distribution of is that
it can be used to provide probability information about the difference
between the sample mean and the population mean.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of : the probability distribution of all possible values of the sample mean .
Relationship Between the Sample Size and the Sampling Distribution of
As the sample size is increased, the standard error of the mean decreases. tttu@hcmiu.edu.vn 13 lOMoARcPSD|364 906 32
As a result, the larger sample size provides a higher probability that the
sample mean is within a specified distance of the population mean.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of point estimator of the population proportion p. tttu@hcmiu.edu.vn 14 lOMoARcPSD|364 906 32 The sampling distribution of tttu@hcmiu.edu.vn 15 lOMoARcPSD|364 906 32
values of the sample proportion . Expected Value of : Standard Deviation of
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Sampling distribution of point estimator of the population proportion p.
Form of the Sampling Distribution of : The sampling distribution of p can be
approximated by a normal distribution whenever: np ≥ 5 and n(1 − p) ≥ 5 tttu@hcmiu.edu.vn 16 lOMoARcPSD|364 906 32
Practical Value of the Sampling Distribution of : can be used to provide
probability information about the difference between the sample proportion
and the population proportion.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Properties of point estimators: three properties of good point
estimators: unbiased, efficiency, and consistency.
Unbiased: The sample statistic is an unbiased estimator of the population parameter
Efficiency: The point estimator with the smaller standard error is said to have
greater relative efficiency than the other. tttu@hcmiu.edu.vn 17 lOMoARcPSD|364 906 32
Consistency: point estimator is consistent if the values of the point estimator
tend to become closer to the population parameter as the sample size becomes
larger. In other words, a large sample size tends to provide a better point
estimate than a small sample size.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Properties of point estimators: three properties of good point
estimators: unbiased, efficiency, and consistency. tttu@hcmiu.edu.vn 18 lOMoARcPSD|364 906 32 Unbiased: tttu@hcmiu.edu.vn 19 lOMoARcPSD|364 906 32
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
•Properties of point estimators: three properties of good point
estimators: unbiased, efficiency, and consistency. tttu@hcmiu.edu.vn 20 lOMoARcPSD|364 906 32 tttu@hcmiu.edu.vn 21 lOMoARcPSD|364 906 32 End of file 1. Any questions? tttu@hcmiu.edu.vn 17
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Other sampling methods: Stratified random sampling Cluster sampling lOMoARcPSD|364 906 32 Systematic sampling Convenience sampling Judgment sampling
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Other sampling methods: Stratified random sampling:
The elements in the population are first divided into groups called strata, such
that each element in the population belongs to one and only one stratum.
The basis for forming the strata, such as department, location, age, industry
type, and so on, is at the discretion of the designer of the sample. tttu@hcmiu.edu.vn 23 lOMoARcPSD|364 906 32
However, the best results are obtained when the elements within each
stratum are as much alike as possible.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Other sampling methods: Cluster sampling:
The elements in the population are first divided into separate groups called clusters.
Each element of the population belongs to one and only one cluster. A simple random
sample of the clusters is then taken.
All elements within each sampled cluster form the sample.
Cluster sampling tends to provide the best results when the elements within the clusters are not alike. tttu@hcmiu.edu.vn 24 lOMoARcPSD|364 906 32
In the ideal case, each cluster is a representative small-scale version of the entire population.
The value of cluster sampling depends on how representative each cluster is of the entire population.
If all clusters are alike in this regard, sampling a small number of clusters will provide
good estimates of the population parameters.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Other sampling methods: Systematic sampling:
In some sampling situations, especially those with large populations, it is time-
consuming to select a simple random sample by first finding a random number tttu@hcmiu.edu.vn 25 lOMoARcPSD|364 906 32
and then counting or searching through the list of the population until the
corresponding element is found.
An alternative to simple random sampling is systematic sampling.
4.4. SAMPLING AND SAMPLING DISTRIBUTIONS •Other sampling methods:
Convenience Sampling: is a nonprobability sampling technique. As the name
implies, the sample is identified primarily by convenience. Elements are included
in the sample without prespecified or known probabilities of being selected.
Judgment sampling: one additional nonprobability sampling technique is
judgment sampling. In this approach, the person most knowledgeable on the
subject of the study selects elements of the population that he or she feels are tttu@hcmiu.edu.vn 26 lOMoARcPSD|364 906 32
most representative of the population. Often this method is a relatively easy way of selecting a sample. tttu@hcmiu.edu.vn 27 lOMoARcPSD|364 906 32 End of file 2. Any questions? tttu@hcmiu.edu.vn 23
EXPLANATION FOR ONLINE MIDTERM EXAM •Date: April 23rd 2020 •Time: 10:30 •Duration: 90 minutes tttu@hcmiu.edu.vn 28 lOMoARcPSD|364 906 32
•Content: Chapters 1, 2, 3, 4
•Structure: 05 questions (mainly focus on calculation, applying equations)
Mean, mode, median, variance, range, quartiles P(A), P(B), P(AUB), P(AՈB), P(A|B), P(B|A), etc. P(X≤a), P(a≤X≤b), etc.
Probability of different distributions
EXPLANATION FOR ONLINE MIDTERM EXAM Rules:
1. Open-book online exam. Please turn on your webcam during the exam time.
2. No talking or sharing materials during the exam.
3. Submission: via Blackboard: Work on this file and upload the file to Blackboard; or via
MS Forms: Solve the equations and submit online via MS Forms (link will be provided)
In case of doing exam via MS Forms: use laptop/computer for full view lOMoARcPSD|364 906 32
(please avoid using cellphone/tablets) Note: Your answers
will be checked via Turnitin for plagiarism. Please WRITE your own answers. tttu@hcmiu.edu.vn 25 tttu@hcmiu.edu.vn 30 lOMoARcPSD|364 906 32 Any questions? Good luck for your exam. tttu@hcmiu.edu.vn 26
Document Outline

• APPLIED STATISTICS
  • Chapter 4: Probability and Distribution
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (1)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (2)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (3)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (4)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (5)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (6)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (7)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (8)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (9)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (10)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (11)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (12)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (13)
    • 4.4. SAMPLING AND SAMPLING DISTRIBUTIONS (14)
    • EXPLANATION FOR ONLINE MIDTERM EXAM