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CSC 323 Quarter: Spring 02/03

CSC 323 Quarter: Spring 02/03. Daniela Stan Raicu School of CTI, DePaul University. Outline. Chapter 5: Sampling Distributions. Population and sample Sampling distribution of a sample mean Central limit theorem Examples. Sample. Population. Introduction.

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CSC 323 Quarter: Spring 02/03

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  1. CSC 323 Quarter: Spring 02/03 Daniela Stan Raicu School of CTI, DePaul University Daniela Stan - CSC323

  2. Outline Chapter 5: Sampling Distributions • Population and sample • Sampling distribution of a sample mean • Central limit theorem • Examples Daniela Stan - CSC323

  3. Sample Population 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 Daniela Stan - CSC323

  4. 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. Daniela Stan - CSC323

  5. 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. Daniela Stan - CSC323

  6. 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. Daniela Stan - CSC323

  7. The Distribution of the Sample Mean Problem: Population: {A,B,C,D,E,F} Population mean:  = .1483 Population Variance:  = .00061 Daniela Stan - CSC323

  8.  = .1483  = .00061 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? Daniela Stan - CSC323

  9. 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 Daniela Stan - CSC323

  10. What is the variance of the sample mean? • An estimator variance reveals a great deal about the quality of the estimator. The variance of thesample 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 Daniela Stan - CSC323

  11. 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? Daniela Stan - CSC323

  12. 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. Daniela Stan - CSC323

  13. The Central Limit Theorem Daniela Stan - CSC323

  14. 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 >= 30 Daniela Stan - CSC323

  15. Importance of the central limit theorem Examples: Daniela Stan - CSC323

  16. is normal Yes Yes No No 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) Daniela Stan - CSC323

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