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Basic Business Statistics (8th Edition) Chapter 7 Sampling Distributions © 2002 Prentice-Hall, Inc. Chap 7-1 Chapter Topics Sampling distribution of the mean Sampling distribution of the proportion Sampling from finite population © 2002 Prentice-Hall, Inc. Chap 7-2 Why Study Sampling Distributions Sample statistics are used to estimate population parameters e.g.: X 50 estimates the population mean Problems: Different samples provide different estimates Large samples give better estimates; large sample costs more How good is the estimate? Approach to solution: Theoretical basis is sampling distribution © 2002 Prentice-Hall, Inc. Chap 7-3 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 © 2002 Prentice-Hall, Inc. Chap 7-4 Developing Sampling Distributions Assume there is a population … Population size N=4 B C Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years A © 2002 Prentice-Hall, Inc. D Chap 7-5 Developing Sampling Distributions (continued) Summary Measures for the Population Distribution N X i 1 P(X) i .3 N 18 20 22 24 21 4 N X i 1 © 2002 Prentice-Hall, Inc. i N .2 .1 0 2 2.236 A B C D (18) (20) (22) (24) X Uniform Distribution Chap 7-6 Developing Sampling Distributions All Possible Samples of Size n=2 1st Obs 2nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 (continued) 16 Sample Means 22 22,18 22,20 22,22 22,24 1st 2nd Observation Obs 18 20 22 24 24 24,18 24,20 24,22 24,24 18 18 19 20 21 16 Samples Taken with Replacement 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 © 2002 Prentice-Hall, Inc. Chap 7-7 Developing Sampling Distributions (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means 1st 2nd Observation Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 © 2002 Prentice-Hall, Inc. P(X) .3 .2 .1 0 _ 18 19 20 21 22 23 24 X Chap 7-8 Developing Sampling Distributions (continued) Summary Measures of Sampling Distribution N X X i 1 N i 18 19 19 16 N X X i 1 i X © 2002 Prentice-Hall, Inc. 21 2 N 18 21 19 21 2 24 16 2 24 21 2 1.58 Chap 7-9 Comparing the Population with its Sampling Distribution Sample Means Distribution n=2 Population N=4 21 2.236 X 21 P(X) .3 P(X) .3 .2 .2 .1 .1 0 0 A B C (18) (20) (22) © 2002 Prentice-Hall, Inc. D X X 1.58 _ 18 19 20 21 22 23 24 X (24) Chap 7-10 Properties of Summary Measures X e.g.: X Is unbiased Standard error (standard deviation) of the sampling distribution X is less than the standard error of other unbiased estimators For sampling with replacement: © 2002 Prentice-Hall, Inc. As n increases, X decreases X n Chap 7-11 Unbiasedness P(X) Unbiased © 2002 Prentice-Hall, Inc. Biased X X Chap 7-12 Less Variability P(X) Sampling Distribution of Median Sampling Distribution of Mean © 2002 Prentice-Hall, Inc. X Chap 7-13 Effect of Large Sample Larger sample size P(X) Smaller sample size © 2002 Prentice-Hall, Inc. X Chap 7-14 When the Population is Normal Population Distribution Central Tendency X Variation X n Sampling with Replacement © 2002 Prentice-Hall, Inc. 10 50 Sampling Distributions n4 n 16 X 5 X 2.5 X 50 X Chap 7-15 When the Population is Not Normal Population Distribution Central Tendency X Variation X n Sampling with Replacement © 2002 Prentice-Hall, Inc. 10 50 Sampling Distributions n4 n 30 X 5 X 1.8 X 50 X Chap 7-16 Central Limit Theorem As Sample Size Gets Large Enough Sampling Distribution Becomes Almost Normal Regardless of Shape of Population X © 2002 Prentice-Hall, Inc. Chap 7-17 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 © 2002 Prentice-Hall, Inc. Chap 7-18 Example: 8 =2 n 25 P 7.8 X 8.2 ? 7.8 8 X X 8.2 8 P 7.8 X 8.2 P X 2 / 25 2 / 25 P .5 Z .5 .3830 Standardized Normal Distribution Sampling Distribution 2 X .4 25 Z 1 .1915 7.8 © 2002 Prentice-Hall, Inc. 8.2 X 8 X 0.5 Z 0 0.5 Z Chap 7-19 Population Proportions Categorical variable p e.g.: Gender, voted for bush, college degree Proportion of population that has a characteristic p Sample proportion provides an estimate X number of successes pS n sample size If two outcomes, X has a binomial distribution Possess or do not possess characteristic © 2002 Prentice-Hall, Inc. Chap 7-20 Sampling Distribution of Sample Proportion Approximated by normal distribution np 5 n 1 p 5 P(ps) .3 .2 .1 0 Mean: p p Sampling Distribution 0 .2 .4 .6 8 1 ps S Standard error: p S © 2002 Prentice-Hall, Inc. p 1 p n p = population proportion Chap 7-21 Standardizing Sampling Distribution of Proportion Z pS pS p S p 1 p n Standardized Normal Distribution Sampling Distribution p pS p Z 1 S p © 2002 Prentice-Hall, Inc. S pS Z 0 Z Chap 7-22 Example: n 200 p .4 P pS .43 ? p .43 .4 S pS P pS .43 P pS .4 1 .4 200 Standardized Normal Distribution Sampling Distribution p Z 1 S © 2002 Prentice-Hall, Inc. P Z .87 .8078 p .43 S pS 0 .87 Z Chap 7-23 Sampling from Finite Sample Modify standard error if sample size (n) is large relative to population size (N ) n .05N or n / N .05 Use finite population correction factor (FPC) Standard error with FPC N n X n N 1 P S © 2002 Prentice-Hall, Inc. p 1 p N n n N 1 Chap 7-24 Chapter Summary Discussed sampling distribution of the sample mean Described sampling distribution of the sample proportion Discussed sampling from finite populations © 2002 Prentice-Hall, Inc. Chap 7-25