CHAPTER TOPICS. Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population

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1 Distribusi Sampel

2 CHAPTER TOPICS Sampling Distribution of the Mean The Central Limit Theorem Sampling Distribution of the Proportion Sampling from Finite Population 2

3 3 WHY STUDY SAMPLING DISTRIBUTIONS Sample Statistics are Used to Estimate Population Parameters E.g., 50 estimates the population mean Problem: Different Samples Provide Different Estimates Large sample gives better estimate; large sample costs more How good is the estimate? Approach to Solution: Theoretical Basis is Sampling Distribution

4 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 4

5 DEVELOPING SAMPLING DISTRIBUTIONS Suppose There is a Population Population Size N=4 Random Variable,, is Age of Individuals B C Values of : 18, 20, 22, 24 Measured in Years A D 5

6 DEVELOPING SAMPLING DISTRIBUTIONS (continued) 6 Summary Measures for the Population Distribution N i i1 N N i1 i N 2 P() A B C D (18) (20) (22) (24) Uniform Distribution

7 Developing Sampling Distributions (continued) 1 st 2 nd Observation Obs ,18 18,20 18,22 18, ,18 20,20 20,22 20, ,18 22,20 22,22 22, ,18 24,20 24,22 24,24 7 ALL POSSIBLE SAMPLES OF SIZE N=2 16 Samples Taken with Replacement 16 Sample Means 1st 2nd Observation Obs

8 Developing Sampling Distributions (continued) SAMPLING DISTRIBUTION OF ALL SAMPLE MEANS 16 Sample Means 1st 2nd Observation Obs P Sample Means Distribution _

9 Developing Sampling Distributions (continued) SUMMARY MEASURES OF SAMPLING DISTRIBUTION N i i1 N 21 N i1 i N

10 .3.2 COMPARING THE POPULATION WITH ITS SAMPLING DISTRIBUTION Population N = 4 Sample Means Distribution n = P.3.2 P A B C D 0 (18) (20) (22) (24) _

11 11 PROPERTIES OF SUMMARY MEASURES I.e., is unbiased Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators For Sampling with Replacement or without Replacement from Large or Infinite Populations: n As n increases, decreases

12 UNBIASEDNESS ( ) f Unbiased Biased 12

13 LESS VARIABILITY Standard Error (Standard Deviation) of the Sampling Distribution is Less Than the Standard Error of Other Unbiased Estimators f Sampling Distribution of Median Sampling Distribution of Mean 13

14 EFFECT OF LARGE SAMPLE For sampling with replacement: A s n increases, decreases f Smaller sample size Larger sample size 14

15 WHEN THE POPULATION IS NORMAL Central Tendency Population Distribution Variation n n Sampling Distributions n

16 WHEN THE POPULATION IS NOT NORMAL Central Tendency Population Distribution Variation n n Sampling Distributions n

17 CENTRAL LIMIT THEOREM As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal Regardless of Shape of Population 17

18 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 Regardless of the Sample Size This is a property of sampling from a normal population distribution and is NOT a result of the central limit theorem 18

19 EAMPLE: P 8 =2 n ? P P 2 / 25 2 / 25 P.5 Z Sampling Distribution Standardized Normal Distribution Z Z Z

20 POPULATION PROPORTIONS p Categorical Variable E.g., Gender, Voted for Bush, College Degree Proportion of Population Having a Characteristic Sample Proportion Provides an Estimate p S n number of successes sample size If Two Outcomes, Has a Binomial Distribution Possess or do not possess characteristic p 20

21 SAMPLING DISTRIBUTION OF SAMPLE PROPORTION Approximated by Normal Distribution np 5 21 n Mean: Standard error: 1 p 5 ps p S p p 1 n p f(p s ) Sampling Distribution p = population proportion p s

22 STANDARDIZING SAMPLING DISTRIBUTION OF PROPORTION Z ps p p S S p p 1 p p S n Sampling Distribution Standardized Normal Distribution 1 p S Z 22 p S p S Z 0 Z

23 EAMPLE: n p P p S ? ps p.43.4 S P ps.43 P P Z p S 200 Sampling Distribution p S Standardized Normal Distribution 1 Z 23 ps p S Z

24 SAMPLING FROM FINITE SAMPLE Modify Standard Error if Sample Size (n) is Large Relative to Population Size (N ) n.05 N or n / N Use Finite Population Correction Factor (fpc) Standard Error with FPC N n n N 1 P S p 1 p N n n N 1

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