5-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

5-2 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Statistics for Business and Economics Chapter 5 Sampling Distributions

5-3 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Content 1.The Concept of a Sampling Distribution 2.Properties of Sampling Distributions: Unbiasedness and Minimum Variance 3.The Sample Distribution of the Sample Mean and the Central Limit Theorem 4.The Sampling Distribution of the Sample Proportion

5-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives Establish that a sample statistic is a random variable with a probability distribution Define a sampling distribution as the probability distribution of a sample statistic Give two important properties of sampling distributions Learn that the sampling distribution of both the sample mean and sample proportion tends to be approximately normal

5-5 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 5.1 The Concept of a Sampling Distribution

5-6 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Parameter & Statistic A parameter is a numerical descriptive measure of a population. Because it is based on all the observations in the population, its value is almost always unknown. A sample statistic is a numerical descriptive measure of a sample. It is calculated from the observations in the sample.

5-7 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Common Statistics & Parameters Sample StatisticPopulation Parameter Variance s2s2  Standard Deviation s  Mean  x  Binomial Proportion pp ^

5-8 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Distribution The sampling distribution of a sample statistic calculated from a sample of n measurements is the probability distribution of the statistic.

5-9 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Developing Sampling Distributions Population size, N = 4 Random variable, x Values of x: 1, 2, 3, 4 Uniform distribution © T/Maker Co. Suppose There’s a Population...

5-10 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Population Characteristics Population DistributionSummary Measure P(x)P(x) x

5-11 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. All Possible Samples of Size n = 2 Sample with replacement Samples 1st Obs 1,11,21,31,4 2,12,22,32,4 3,13,23,33,4 4,14,24,34,4 2nd Observation st Obs 16 Sample Means

5-12 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Distribution of All Sample Means nd Observation st Obs 16 Sample MeansSampling Distribution of the Sample Mean P(x) x

5-13 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Summary Measure of All Sample Means

5-14 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Comparison PopulationSampling Distribution P(x)P(x) x P(x) x

5-15 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 5.2 Properties of Sampling Distributions: Unbiasedness and Minimum Variance

5-16 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Point Estimator A point estimator of a population parameter is a rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the population parameter.

5-17 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Estimates If the sampling distribution of a sample statistic has a mean equal to the population parameter the statistic is intended to estimate, the statistic is said to be an unbiased estimate of the parameter. If the mean of the sampling distribution is not equal to the parameter, the statistic is said to be a biased estimate of the parameter.

5-18 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Example Probability Distribution x023 p(x)p(x)  = = = 1.247

5-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Example Sampling Distribution of for n = 2 (3 possible samples, each with a sample mean) 1 p(x)p(x) E( ) = is the same as . is an unbiased estimator of .

5-20 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 5.3 The Sampling Distribution of a Sample Mean and the Central Limit Theorem

5-21 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Properties of the Sampling Distribution of x 1.Mean of the sampling distribution equals mean of sampled population*, that is, 2.Standard deviation of the sampling distribution equals That is,

5-22 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Standard Error of the Mean The standard deviation is often referred to as the standard error of the mean.

5-23 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Theorem 5.1 If a random sample of n observations is selected from a population with a normal distribution, the sampling distribution of will be a normal distribution.

5-24 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling from Normal Populations Central Tendency Dispersion –Sampling with replacement n =16   x = 2.5  = 50  = 10 x n = 4   x = 5  x = 50 - x Sampling Distribution Population Distribution

5-25 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Standardizing the Sampling Distribution of x Standardized Normal Distribution  = 0  = 1 z Sampling Distribution x  x  x

5-26 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Thinking Challenge You’re an operations analyst for AT&T. Long- distance telephone calls are normally distributed with  = 8 min. and  = 2 min. If you select random samples of 25 calls, what percentage of the sample means would be between 7.8 & 8.2 minutes? © T/Maker Co.

5-27 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling Distribution Solution* Sampling Distribution 8   x =  x 0  = 1 –.50 z Standardized Normal Distribution.1915

5-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sampling from Non-Normal Populations Central Tendency Dispersion –Sampling with replacement Population Distribution Sampling Distribution n =30   x = 1.8 n = 4   x = 5  = 50  = 10 x  x = 50 - x

5-29 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Limit Theorem Consider a random sample of n observations selected from a population (any probability distribution) with mean μ and standard deviation . Then, when n is sufficiently large, the sampling distribution of will be approximately a normal distribution with mean and standard deviation The larger the sample size, the better will be the normal approximation to the sampling distribution of.

5-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Limit Theorem x As sample size gets large enough (n  30)... sampling distribution becomes almost normal.

5-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Limit Theorem Example The amount of soda in cans of a particular brand has a mean of 12 oz and a standard deviation of.2 oz. If you select random samples of 50 cans, what percentage of the sample means would be less than oz? SODA

5-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Central Limit Theorem Solution* Sampling Distribution 12   x =  x 0  = 1 –1.77 z.0384 Standardized Normal Distribution.4616 Shaded area exaggerated

5-33 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 5.4 The Sampling Distribution of the Sample Proportion

5-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Sample Proportion Just as the sample mean is a good estimator of the population mean, the sample proportion—denoted — is a good estimator of the population proportion p. How good the estimator is will depend on the sampling distribution of the statistic. This sampling distribution has properties similar to those of the sampling distribution of

5-35 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Properties of Probability Distributions Sampling distribution of a statistic—the theoretical probability distribution of the statistic in repeated sampling Unbiased estimator—a statistic with a sampling distribution mean equal to the population parameter being estimated

5-36 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Properties of Probability Distributions Central Limit Theorem—the sampling distribution of the sample mean,, or the sample proportion,, is approximately normal for large n is the minimum-variance unbiased estimator (MVUE) is the MVUE of p

5-37 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Key Ideas Generating the Sampling Distribution of x