CSC 323 Quarter: Winter

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CSC 323 Quarter Winter 02/03

• Daniela Stan Raicu
• School of CTI, DePaul University

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Outline
Chapter 5 Sampling Distributions

• Population and sample
• Sampling distribution of a sample mean
• Central limit theorem
• Examples

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Introduction

• This chapter begins a bridge from the study of
  probabilities to the study of statistical
  inference, by introducing the sampling
  distribution.
• Quality of sample data

• The quality of all statistical
• analysis depends on the quality
• of the sample data
• If the data sample is not representative,
  analyzing the data and drawing conclusions will
  be unproductive-at best.

Random Sampling every unit in the population
has an equal chance to be chosen
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Some definitions

• Parameter A number describing a population.
• Statistic A number describing a sample.

1. A random sample should represent the
population well, so sample statistics from a
random sample should provide reasonable estimates
of population parameters.
Sample statistics Population parameter
Sample mean x ?
Sample proportion p_hat p
Sample variance s2 ?2
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Some definitions (cont.)
2. All sample statistics have some error in
estimating population parameters.
3. If repeated samples are taken from a
population and the same statistic (e.g. mean) is
calculated from each sample, the statistics will
vary, that is, they will have a distribution.
4. A larger sample provides more information than
a smaller sample so a statistic from a large
sample should have less error than a statistic
from a small sample.
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Describing the Sample Mean

• Let us assume that we want to estimate the mean
  ? of the population since usually this is the
  first piece of information that an analyst wants
  to analyze

• Since the value of the sample mean depends on the
  particular sample we draw, the sample mean is a
  variable with a huge number of possible values.
• The sample mean is a random variable because the
  samples are drawn randomly.
• The best way to summarize this vast amount of
  information is to describe it with a probability
  distribution.

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The Distribution of the Sample Mean
Problem
Population A,B,C,D,E,F
Population mean ? .1483
Population Variance ? .00061
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The Distribution of the Sample Mean
Assumptions

• What is the central value of the variable x?
• What is its variability?
• Is there a familiar pattern in the variability?

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What is the central value of the sample mean?

• For large samples, the distribution of x should
  be symmetrical x should be larger than ? about
  50 of the time and x should be smaller than ?
  about 50 of the time.

It can be shown theoretically (Central Limit
theorem) that the mean of the sample means equals
the population mean E(x) ?
In our example, E(x) 0.1483 ?
x is an unbiased estimator
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What is the variance of the sample mean?

• An estimator variance reveals a great deal about
  the quality of the estimator.

The variance of the sample mean s2 ?2/n Where
?2 variance of the population n sample size
Increase of the sample size n
Decrease of the variance s2
Better accuracy of the estimator
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Accuracy of the Estimator
As in many problems, there is a trade off between
accuracy and dollars.
What we will get from our money if we
invest dollars in obtaining a larger size?
n 100? n 200?
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Is there a familiar pattern in the data?

• As the sample size becomes larger, the
  distribution of the sample mean becomes closer to
  a normal distribution, regardless the
  distribution of the population from which the
  sample is drawn.

• The central limit theorem summarizes the
  distribution of the
• sample mean.

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The Central Limit Theorem
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Importance of the central limit theorem

• The most important feature is that it can be
  applied to
• any population as long as the sample size n is
  large enough.

How large is large? n gt 30
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Importance of the central limit theorem
Examples
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Is x normal distributed?
Is the population normal?
Yes
No
Is ?
Is ?
may or may not be considered normal
has t-student distribution
is considered to be normal
(We need more info)