Behavioral Statistics Sampling Distributions Chapter 6

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 Probability Problems Involving Sampling Distributions As a result of this class, you should be able to ...

Inferential Statistics 9

Statistical Methods

Inferential Statistics 1. Involves: Estimation Hypothesis Testing 2. Purpose Make Decisions about Population Characteristics Population?

Inference Process

Inference Process Population

Inference Process Population Sample

Inference Process Population Sample statistic (X) Sample

Inference Process Estimates & tests Population Sample statistic (X)

Estimators 1. Random Variables Used to Estimate a Population Parameter 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  3. Theoretical Basis Is Sampling Distribution

Sampling Distributions 9

Sampling Distribution 1. Theoretical Probability Distribution 2. Random Variable is Sample Statistic 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

Developing Sampling Distributions Suppose There’s a Population ... Population Size, N = 4 Random Variable, x, Is # Errors in Work Values of x: 1, 2, 3, 4 Uniform Distribution © 1984-1994 T/Maker Co.

Population Characteristics Summary Measures Population Distribution Have students verify these numbers.

All Possible Samples of Size n = 2 Sample with replacement

All Possible Samples of Size n = 2 16 Sample Means Sample with replacement

Sampling Distribution of All Sample Means

Summary Measures of All Sample Means Have students verify these numbers.

Sampling Distribution Comparison Population Sampling Distribution

Standard Error of Mean 1. Standard Deviation of All Possible Sample Means,X Measures Scatter in All Sample Means,X 2. Less Than Pop. Standard Deviation

Standard Error of Mean 1. Standard Deviation of All Possible Sample Means,X Measures Scatter in All Sample Means,X 2. Less Than Pop. Standard Deviation 3. Formula (Sampling With Replacement)

Properties of Sampling Distribution of Mean 9

Properties of Sampling Distribution of Mean 1. Unbiasedness Mean of Sampling Distribution Equals Population Mean 2. Efficiency 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 An estimator is a random variable used to estimate a population parameter (characteristic). Unbiasedness An estimator is unbiased if the mean of its sampling distribution is equal to the population parameter. Efficiency The efficiency of an unbiased estimator is measured by the variance of its sampling distribution. If two estimators, with the same sample size, are both unbiased, then the one with the smaller variance has greater relative efficiency. Consistency An estimator is a consistent estimator of a population parameter if the larger the sample size, the more likely it is that the estimate will come close to the parameter.

Unbiasedness Unbiased Biased 

Sampling distribution of mean Sampling distribution of median Efficiency Sampling distribution of mean Sampling distribution of median 

Consistency Larger sample size Smaller sample size 

Sampling from Normal Populations 9

Sampling from Normal Populations Central Tendency Dispersion Sampling with replacement Population Distribution Sampling Distribution n = 4 X = 5 n =16 X = 2.5

Standardizing Sampling Distribution of Mean Standardized Normal Distribution

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.

Sampling Distribution Solution* Standardized Normal Distribution .3830 .1915 .1915

Sampling from Non-Normal Populations 9

Sampling from Non-Normal Populations Central Tendency Dispersion Sampling with replacement Population Distribution Sampling Distribution n = 4 X = 5 n =30 X = 1.8

Central Limit Theorem 9

Central Limit Theorem

Central Limit Theorem As sample size gets large enough (n  30) ...

Central Limit Theorem As sample size gets large enough (n  30) ... sampling distribution becomes almost normal.

Central Limit Theorem As sample size gets large enough (n  30) ... sampling distribution becomes almost normal.

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 Probability Problems Involving Sampling Distributions

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