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06/19/22 Daniela Stan - CSC323 1 CSC 323 CSC 323 Quarter: Quarter: Spring 02/03 Spring 02/03 Daniela Stan Raicu School of CTI, DePaul University
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CSC 323 CSC 323 Quarter:Quarter: Spring 02/03 Spring 02/03
Daniela Stan RaicuSchool of CTI, DePaul University
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OutlineOutline
Population and sample Sampling distribution of a sample mean Central limit theorem Examples
Chapter 5: Sampling Distributions
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IntroductionIntroduction
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:
Sample
Population
• 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 definitionsSome 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 Some definitions (cont.)(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 MeanDescribing 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 MeanThe Distribution of the Sample MeanProblem:
Population:{A,B,C,D,E,F}
Population mean: = .1483
Population Variance: = .00061
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The Distribution of the Sample MeanThe Distribution of the Sample MeanAssumptions:
= .1483
= .00061
• 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 What is the central value of the sample mean?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 What is the variance of the sample mean?sample mean?
• An estimator variance reveals a great deal about the quality of the estimator.
The variance of the sample mean
s2 = 2/nWhere 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 EstimatorAccuracy of the Estimator
As in many problems, thereis a trade off between accuracy and dollars.
What we will get from our money if we investdollars in obtaining a larger size?
n = 100?n = 200?
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Is there a familiar pattern Is there a familiar pattern in the data?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 TheoremThe Central Limit Theorem
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Importance of the Importance of the central limit theorem 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 >= 30
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Importance of the Importance of the central limit theorem central limit theorem
Examples:
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Is the population normal?
may or may not be
considered normal
is normalX is considered to be normal
X
Is ?30n
X
(We need more info)
Yes
Yes
No
No
Is ?30n
has t-student
distribution
X
Yes No
Is x normal distributed?Is x normal distributed?
