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BA 201 Lecture 11 Sampling Distributions © 2001 Prentice-Hall, Inc. Chap 7-1 Topics Estimation Process Point Estimates Interval Estimates Sampling Distribution of the Mean © 2001 Prentice-Hall, Inc. Chap 7-2 Population and Sample Population p.?? Sample Use statistics to summarize features Use parameters to summarize features Inference on the population from the sample © 2001 Prentice-Hall, Inc. Chap 7-3 pp.?? Estimation Process Population Mean, , is unknown Random Sample X 50 I conjecture that the population mean, , is 50 Sample © 2001 Prentice-Hall, Inc. Chap 7-4 p.267 Point Estimates Estimate Population Parameters … Mean Proportion Variance Difference © 2001 Prentice-Hall, Inc. p with Sample Statistics X PS 1 2 2 S 2 X1 X 2 Chap 7-5 Another Point Estimate Here is a link to some of the most recent poll results © 2001 Prentice-Hall, Inc. Chap 7-6 p.? Drawback of Point Estimates Q. What is the probability that a point estimate will equal to the true parameter that is being estimated? A. Zero. Theoretically, you will never obtain a point estimate that equals the unknown parameter. © 2001 Prentice-Hall, Inc. Chap 7-7 pp.?? Interval Estimation Process Population Mean, , is unknown Random Sample X 50 I am 95% confident that is between 40 & 60. Sample © 2001 Prentice-Hall, Inc. Chap 7-8 p.267 Interval Estimates Provides Range of Values Take into consideration variation in sample statistics from sample to sample Based on observation from 1 sample Give Information about Closeness to Unknown Population Parameters Stated in terms of level of confidence © 2001 Prentice-Hall, Inc. Never 100% sure Chap 7-9 pp.?? Confidence Interval Estimates Confidence Intervals Mean Known © 2001 Prentice-Hall, Inc. Proportion Unknown Chap 7-10 Why Study Sampling Distributions Sample Statistics are Used to Estimate Population Parameters E.g. X 50 estimates the population mean X Problems: Different Sample Provides Different Estimate p.252 Large sample gives better estimate; large sample costs more How good is the estimate? Approach to Solution: Theoretical Basis is Sampling Distribution © 2001 Prentice-Hall, Inc. Chap 7-11 p.252 Sampling Distribution Theoretical Probability Distribution of a Sample Statistic Sample Statistic is a Random Variable Sample mean, sample proportion Results from Taking All Possible Samples of the Same Size © 2001 Prentice-Hall, Inc. Chap 7-12 pp. 256-261 When the Population is Normal Central Tendency Population Distribution X 10 X X Variation X X n Sampling with Replacement © 2001 Prentice-Hall, Inc. X 50 Sampling Distributions n4 X 5 n 16 X 2.5 X 50 X Chap 7-13 When the Population is Not pp.261-265 Normal Population Distribution Central Tendency X Variation X n Sampling with Replacement © 2001 Prentice-Hall, Inc. X 10 X 50 Sampling Distributions n4 n 30 X 5 X 1.8 X 50 X Chap 7-14 p.261 Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X © 2001 Prentice-Hall, Inc. Chap 7-15 Applet to Illustrate the CLT Click here to access the applet that will illustrate the Central Limit Theorem in action. © 2001 Prentice-Hall, Inc. Chap 7-16 p.265 How Large is Large Enough? For Most Distributions, n>30 For Fairly Symmetric Distributions, n>15 For Normal Distribution, the Sampling Distribution of the Mean is Always Normally Distributed This is a property of sampling from a normal population distribution and is NOT a result of the central limit theorem © 2001 Prentice-Hall, Inc. Chap 7-17 Summary Illustrated Estimation Process Discussed Point Estimates Addressed Interval Estimates Discussed Sampling Distribution of the Sample Mean © 2001 Prentice-Hall, Inc. Chap 7-18