Basic Business Statistics (8th Edition) Chapter 7 Sampling Distributions © 2002 Prentice-Hall, Inc.

Why Study Sampling Distributions Sample statistics are used to estimate population parameters e.g.: 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.

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.

Developing Sampling Distributions Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 measured in years C B D A © 2002 Prentice-Hall, Inc.

Developing Sampling Distributions (continued) Summary Measures for the Population Distribution P(X) .3 .2 .1 X A B C D (18) (20) (22) (24) Uniform Distribution © 2002 Prentice-Hall, Inc.

All Possible Samples of Size n=2 Developing Sampling Distributions (continued) All Possible Samples of Size n=2 16 Sample Means 16 Samples Taken with Replacement © 2002 Prentice-Hall, Inc.

Sampling Distribution of All Sample Means Developing Sampling Distributions (continued) Sampling Distribution of All Sample Means Sample Means Distribution 16 Sample Means P(X) .3 .2 .1 _ X 18 19 20 21 22 23 24 © 2002 Prentice-Hall, Inc.

Summary Measures of Sampling Distribution Developing Sampling Distributions (continued) Summary Measures of Sampling Distribution © 2002 Prentice-Hall, Inc.

Comparing the Population with its Sampling Distribution Sample Means Distribution n = 2 Population N = 4 P(X) P(X) .3 .3 .2 .2 .1 .1 _ X A B C D (18) (20) (22) (24) 18 19 20 21 22 23 24 X © 2002 Prentice-Hall, Inc.

Properties of Summary Measures e.g.: Is unbiased Standard error (standard deviation) of the sampling distribution is less than the standard error of other unbiased estimators For sampling with replacement: As n increases, decreases © 2002 Prentice-Hall, Inc.

When the Population is Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement © 2002 Prentice-Hall, Inc.

When the Population is Not Normal Population Distribution Central Tendency Variation Sampling Distributions Sampling with Replacement © 2002 Prentice-Hall, Inc.

Central Limit Theorem Sampling Distribution Becomes Almost Normal Regardless of Shape of Population As Sample Size Gets Large Enough © 2002 Prentice-Hall, Inc.

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.

Standardized Normal Distribution Example: Standardized Normal Distribution Sampling Distribution © 2002 Prentice-Hall, Inc.

Population Proportions Categorical variable e.g.: Gender, voted for bush, college degree Proportion of population that has a characteristic Sample proportion provides an estimate If two outcomes, X has a binomial distribution Possess or do not possess characteristic © 2002 Prentice-Hall, Inc.

Sampling Distribution of Sample Proportion Approximated by normal distribution Mean: Standard error: Sampling Distribution P(ps) .3 .2 .1 ps 0 . 2 .4 .6 8 1 p = population proportion © 2002 Prentice-Hall, Inc.

Standardizing Sampling Distribution of Proportion Standardized Normal Distribution Sampling Distribution © 2002 Prentice-Hall, Inc.

Standardized Normal Distribution Example: Standardized Normal Distribution Sampling Distribution © 2002 Prentice-Hall, Inc.

Sampling from Finite Sample Modify standard error if sample size (n) is large relative to population size (N ) Use finite population correction factor (FPC) Standard error with FPC © 2002 Prentice-Hall, Inc.