PROBABILITY AND STATISTICS
Lab, Seminar, Lecture 4.

Behavior of the sample average

X-bar
The topic of 4th seminar&lab is the average of the

population that has a certain characteristic. This average is the population parameter of interest, denoted by the greek letter mu. We estimate this parameter with the statistic x-bar, the average in the sample.

Probability and statistics - Karol Flisikowski

X-bar Definition

1 x xi n i 1
Probability and statistics - Karol Flisikowski

n

Sampling Distribution of x-bar
How does x-bar behave? To study the behavior,

imagine taking many random samples of size n, and computing an x-bar for each of the samples. Then we plot this set of x-bars with a histogram.

Probability and statistics - Karol Flisikowski

Sampling Distribution of x-bar

Probability and statistics - Karol Flisikowski

Central Limit Theorem
The key to the behavior of x-bar is the central limit

theorem. It says: Suppose the population has mean, m, and standard deviation s. Then, if the sample size, n, is large enough, the distribution of the sample mean, x-bar will have a normal shape, the center will be the mean of the original population, m, and the standard deviation of the x-bars will be s divided by the square root of n. Probability and statistics - Karol Flisikowski

Central Limit Theorem
If the CLT holds we have,

Normal shape
Center = mu

Spread = sigma/sqroot n.

Probability and statistics - Karol Flisikowski

When Does CLT Hold?
Answer generally depends on the sample size, n,

and the shape of the original distribution. General Rule: the more skewed the population distribution of the data, the larger sample size is needed for the CLT to hold.

Probability and statistics - Karol Flisikowski

CLT

Previous overhead shows the original population distribution in (a),

and increasing sample sizes through graphs (b), (c), and (d). Notice that it takes...

...SAMPLINGSAMPLING
SAMPLINGDISTRIBUTION
(THEORETICAL)
SAMPLING TECHNIQUES (APPLIED)
8/13/2014
QM_Session 14 15
SAMPLING TERMS
An unit/element is the entity on which data are collected.
A population is a collection of all the units/elements of
interest.
A sample is a subset of the population.
The sampled population is the population from
which the sample is drawn.
A frame is a list of the elements/units that the sample will
be selected from.
8/13/2014
QM_Session 14 15
Parameter and Statistic
Parameter is a
population
characteristic
Eg. µ , P, σ
Statistic is a sample
characteristic
Eg. x , s, p
Using Sample
• Statistical Inference:
On basis of sample statistics
derived from limited and
incomplete sample
information
Predict and forecast values of
population parameters...
Estimate and test hypotheses
about values of population
parameters...
Make decisions...
Make generalizations
about the
characteristics of a
population...
8/13/2014
On the basis of
observations of a
sample, a part of a
population
QM_Session 14 15
Selecting a SampleSampling from a Finite Population
Sampling from an Infinite Population
8/13/2014
QM_Session 14 15
Sampling from a Finite Population
Finite...

...Chapter 7
Sampling and SamplingDistributions
6-1
Learning Objectives
In this chapter, you learn:
The concept of the samplingdistribution
To compute probabilities related to the samplemean and the sample proportion
The importance of the Central Limit Theorem
To distinguish between different survey
sampling methods
To evaluate survey worthiness and survey errors
7-2
Reasons for Drawing a Sample
Selecting a sample is less time-consuming than
selecting every item in the population (census).
Selecting a sample is less costly than selecting
every item in the population.
An analysis of a sample is less cumbersome
and more practical than an analysis of the
entire population.
7-3
A Sampling Process Begins With A
Sampling Frame
The sampling frame is a listing of items that
make up the population
Frames are data sources such as population
lists, directories, or maps
Inaccurate or biased results can result if a
frame excludes certain portions of the
population
Using different frames to generate data can
lead to dissimilar conclusions
7-4
Types of Samples Used
Nonprobability Sample
Items included are chosen without regard to
their probability of occurrence
Probability Sample
...

...Samplingdistribution
The samplingdistribution is the distribution of the values of a sample statistic computed for each possible sample that could be drawn from the target population under a specified sampling plan. Because many different samples could be drawn from a population of elements, the sample statistics derived from any onesample will likely not equal the population parameters. As a result, the samplingdistribution supplies an approximation of the true value’s population parameters.
The main properties of the samplingdistribution are:
1. Normally distributed for large samples.
2. The mean of the samplingdistribution of the mean equals , the population parameter for the mean.
3. The standard error of the mean is the standard deviation of the samplingdistribution.
4. Assuming no measurement error, the reliability of an estimate of a population parameter can be assessed in terms of its standard error.
5. The standard error of the mean can be estimated by using the sample standard deviation, s, as an estimator of .
6. z values calculate the area under the sampling...

...12
SAMPLING MECHANICS
Sampling is an activity that involves the selection of individual people, data or things, from a target population/universe.
A population, or universe, is the entire set people data or things that is the subject of exploration.
A census involves obtaining information, not from a sample, but rather from the entire population or universe.
A sample (as opposed sampling) is a subset of the population/universe.
For Marketing Research purposes, sampling usually involves people, not data or things.
Sampling Plans are strategies and mechanics for selecting members of the sample from the population:
1. Define the population. It is usually limited based on some set of characteristics, e.g., males, aged 21-39, who have consumed alcoholic beverages within the past 3 months for a beer study.
2. Choose data collection methodology. What kind of information do you require from the sample, how will they be identified, where are they available, etc.
3. Set sampling frame. This is as exhaustive a list as operationally and economically possible that represents the population and is also accessible utilizing the selected methodology.
4. Choose sampling method.
• Probability samples are those that allow all members of the sampling frame an equal...

...HOMEWORK 2
FROM CHAPTER 6 and 7, NORMAL DISTRIBUTION AND SAMPLING
Instructor: Asiye Aydilek
PART 1- Multiple Choice Questions
____ 1. For the standard normal probability distribution, the area to the left of the mean is
a.
–0.5
c.
any value between 0 to 1
b.
0.5
d.
1
Answer: B
The total area under the curve is 1. The area on the left is the half of 1 which is 0.5.
____ 2. Which of the following is not a characteristic of the normal probability distribution?
a.
The mean and median are equal
b.
The mean of the distribution can be negative, zero, or positive
c.
The distribution is symmetrical
d.
The standard deviation must be 1
o
Answer:D
Normal distribution is symmetric. So, mean=median
The mean could be any number.
The standard deviation is positive but it is not always 1.
____ 3. Larger values of the standard deviation result in a normal curve that is
a.
shifted to the right
c.
narrower and more peaked
b.
shifted to the left
d.
wider and flatter
Answer: D
The Total area under the curve is 1. If the standard deviation is larger, it means the curve is wider, but the height is lower.
____ 4. Which of the following is not a characteristic of the normal probability distribution?
a.
Symmetry
b.
The total area under the curve is always equal to...

...SAMPLINGDISTRIBUTIONS
|6.1 POPULATION AND SAMPLINGDISTRIBUTION |
|6.1.1 Population Distribution |
Suppose there are only five students in an advanced statistics class and the midterm scores of these five students are:
70 78 80 80 95
Let x denote the score of a student.
• Mean for Population
Based on Example 1, to calculate mean for population:
[pic]
• Standard Deviation for Population
Based on example 1, to calculate standard deviation for population:
[pic]
|6.1.2 SamplingDistribution |
▪ Sample statistic such as median, mode, mean and standard deviation
6.1.2.1 The SamplingDistribution of the SampleMean
Reconsider the population of midterm scores of five students given in example 1. Let say we draw all possible samples of three numbers each...

...Simple random sample (SRS)
In statistics, a simple random sample from a population is a sample chosen randomly, so that each possible sample has the same probability of being chosen. One consequence is that each member of the population has the same probability of being chosen as any other. In small populations such sampling is typically done "without replacement", i.e., one deliberately avoids choosing any member of the population more than once. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement.
Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete sampling frame, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population.
Advantages are that it is free of classification error, and it requires minimum advance knowledge of the population. It best suits situations where the population is fairly homogeneous and not much information is available about the population. If these conditions are not true, stratified sampling may be a better choice....

...Sampling and SamplingDistributions
7-1
Learning Objectives
In this chapter, you learn:
To distinguish between different sampling methods The concept of the samplingdistribution To compute probabilities related to the samplemean and the sample proportion The importance of the Central Limit Theorem
7-2
Why Sample?
DCOVA
Selecting a sample is less time-consuming than selecting every item in the population (census).
An analysis of a sample is less cumbersome and more practical than an analysis of the entire population.
7-3
A Sampling Process Begins With A Sampling Frame DCOVA
The sampling frame is a listing of items that make up the population Frames are data sources such as population lists, directories, or maps Inaccurate or biased results can result if a frame excludes certain portions of the population Using different frames to generate data can lead to dissimilar conclusions
7-4
Types of SamplesSamples
DCOVA
Non-Probability Samples
Probability Samples
Judgment
Convenience
Simple Random
Stratified Cluster
Systematic
7-5
DCOVA
In a nonprobability sample, items included are chosen without regard to their probability of...

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