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6 - 1 © 1998 Prentice-Hall, Inc. Chapter 6 Sampling Distributions
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6 - 2 © 1998 Prentice-Hall, Inc. Learning Objectives 1.Describe the properties of estimators 2.Explain sampling distribution 3.Describe the relationship between populations & sampling distributions 4.State the Central Limit Theorem 5.Solve a probability problem involving sampling distributions
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6 - 3 © 1998 Prentice-Hall, Inc. Inferential Statistics
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6 - 4 © 1998 Prentice-Hall, Inc. Types of Statistical Applications
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6 - 5 © 1998 Prentice-Hall, Inc. Inferential Statistics 1.Involves Estimation Estimation Hypothesis testing Hypothesis testing
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6 - 6 © 1998 Prentice-Hall, Inc. Inferential Statistics 1.Involves Estimation Estimation Hypothesis testing Hypothesis testing Population?
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6 - 7 © 1998 Prentice-Hall, Inc. Inferential Statistics 1.Involves Estimation Estimation Hypothesis testing Hypothesis testing 2.Purpose Make decisions about population characteristics Make decisions about population characteristics Population?
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6 - 8 © 1998 Prentice-Hall, Inc. Inference Process
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6 - 9 © 1998 Prentice-Hall, Inc. Inference Process Population
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6 - 10 © 1998 Prentice-Hall, Inc. Inference Process Population Sample
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6 - 11 © 1998 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X)
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6 - 12 © 1998 Prentice-Hall, Inc. Inference Process Population Sample Sample statistic (X) Estimate & test population parameter
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6 - 13 © 1998 Prentice-Hall, Inc. 1.Random variables used to estimate a population parameter Sample mean, sample proportion, sample median Sample mean, sample proportion, sample median 2.Example: Sample mean x is an estimator of population mean If x = 3 then 3 is the estimate of If x = 3 then 3 is the estimate of 3.Theoretical basis is sampling distribution Estimators
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6 - 14 © 1998 Prentice-Hall, Inc. Sampling Distributions
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6 - 15 © 1998 Prentice-Hall, Inc. 1.Theoretical probability distribution 2.Random variable is sample statistic Sample mean, sample proportion etc. Sample mean, sample proportion etc. 3.Results from drawing all possible samples of a fixed size 4.List of all possible [ x, P( x) ] pairs Sampling distribution of mean Sampling distribution of mean Sampling Distribution
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6 - 16 © 1998 Prentice-Hall, Inc. Developing Sampling Distributions Population size, N = 4 Random variable, x, is # televisions owned Values of x: 1, 2, 3, 4 Equally distributed (p=1/4) © 1984-1994 T/Maker Co. Suppose there’s a population...
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6 - 17 © 1998 Prentice-Hall, Inc. Population Characteristics
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6 - 18 © 1998 Prentice-Hall, Inc. Population Characteristics Summary Measures
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6 - 19 © 1998 Prentice-Hall, Inc. Population Characteristics Summary Measures
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6 - 20 © 1998 Prentice-Hall, Inc. Population Characteristics Population Distribution Summary Measures
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6 - 21 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n = 2
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6 - 22 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n = 2 16 Samples Sample with replacement
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6 - 23 © 1998 Prentice-Hall, Inc. Let’s Draw All Possible Samples of Size n=2 16 Samples 16 Sample Means Sample with replacement
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6 - 24 © 1998 Prentice-Hall, Inc. Sampling Distribution of All Sample Means 16 Sample Means Sampling Distribution
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6 - 25 © 1998 Prentice-Hall, Inc. Summary Measures of All Sample Means (n=16)
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6 - 26 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution
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6 - 27 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution Population
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6 - 28 © 1998 Prentice-Hall, Inc. Comparison of Population & Sampling Distribution Population Sampling Distribution
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6 - 29 © 1998 Prentice-Hall, Inc. Standard Error of Mean 1.Standard deviation of all possible sample means, x Measures scatter in all sample means, x Measures scatter in all sample means, x 2.Less than pop. standard deviation
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6 - 30 © 1998 Prentice-Hall, Inc. Standard Error of Mean 1.Standard deviation of all possible sample means, x Measures scatter in all sample means, x Measures scatter in all sample means, x 2.Less than pop. standard deviation 3.Formula (sampling with replacement)
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6 - 31 © 1998 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean
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6 - 32 © 1998 Prentice-Hall, Inc. Properties of Sampling Distribution of Mean 1.Unbiasedness Mean of sampling distribution equals population mean Mean of sampling distribution equals population mean 2.Efficiency Sample mean comes closer to population mean than any other unbiased estimator Sample mean comes closer to population mean than any other unbiased estimator 3.Consistency As sample size increases, variation of sample mean from population mean decreases As sample size increases, variation of sample mean from population mean decreases
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6 - 33 © 1998 Prentice-Hall, Inc. Unbiasedness UnbiasedBiased
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6 - 34 © 1998 Prentice-Hall, Inc. Efficiency Sampling distribution of median Sampling distribution of mean
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6 - 35 © 1998 Prentice-Hall, Inc. Consistency Smaller sample size Larger sample size
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6 - 36 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations
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6 - 37 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency
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6 - 38 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency
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6 - 39 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement
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6 - 40 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement Population Distribution
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6 - 41 © 1998 Prentice-Hall, Inc. Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Central Tendency Dispersion Sampling with replacement Population Distribution Sampling Distribution n =16 X = 2.5 n = 4 X = 5
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6 - 42 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Suppose you want to make probability statements about the sampling distribution...
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6 - 43 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution
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6 - 44 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution Standardized Normal Distribution
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6 - 45 © 1998 Prentice-Hall, Inc. Standardizing Sampling Distribution of Mean Sampling Distribution Standardized Normal Distribution
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6 - 46 © 1998 Prentice-Hall, Inc. Thinking Challenge You’re an operations analyst for AT&T. Long- distance telephone calls are normally distribution with = 8 min. & = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? © 1984-1994 T/Maker Co. AloneGroupClass
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6 - 47 © 1998 Prentice-Hall, Inc. Sampling Distribution Solution* Sampling Distribution.3830.3830.1915.1915 Standardized Normal Distribution
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6 - 48 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations
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6 - 49 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency
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6 - 50 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency
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6 - 51 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency Dispersion Sampling with replacement Sampling with replacement Central Tendency Dispersion Sampling with replacement Sampling with replacement
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6 - 52 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency Dispersion Sampling with replacement Sampling with replacement Central Tendency Dispersion Sampling with replacement Sampling with replacement Population Distribution
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6 - 53 © 1998 Prentice-Hall, Inc. Sampling from Non-Normal Populations Central Tendency Dispersion Sampling with replacement Sampling with replacement Central Tendency Dispersion Sampling with replacement Sampling with replacement Population Distribution Sampling Distribution n =30 X = 1.8 n = 4 X = 5
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6 - 54 © 1998 Prentice-Hall, Inc. Central Limit Theorem
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6 - 55 © 1998 Prentice-Hall, Inc. Central Limit Theorem
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6 - 56 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n 30)...
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6 - 57 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n 30)... sampling distribution becomes almost normal.
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6 - 58 © 1998 Prentice-Hall, Inc. Central Limit Theorem As sample size gets large enough (n 30)... sampling distribution becomes almost normal.
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6 - 59 © 1998 Prentice-Hall, Inc. Conclusion 1.Described the properties of estimators 2.Explained sampling distribution 3.Described the relationship between populations & sampling distributions 4.Stated the Central Limit Theorem 5.Solved a probability problem involving sampling distributions
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6 - 60 © 1998 Prentice-Hall, Inc. This Class... 1.What was the most important thing you learned in class today? 2.What do you still have questions about? 3.How can today’s class be improved? Please take a moment to answer the following questions in writing:
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