Chapter 6 Sampling and Sampling Distributions Statistics for Business and Economics 8th Edition Chapter 6 Sampling and Sampling Distributions Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distributions of Sample Means 6.2 Sampling Distributions Sampling Distributions of Sample Means Sampling Distributions of Sample Proportions Sampling Distributions of Sample Variances Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Sample Mean Let X1, X2, . . ., Xn represent a random sample from a population The sample mean value of these observations is defined as Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Standard Error of the Mean Different samples of the same size from the same population will yield different sample means A measure of the variability in the mean from sample to sample is given by the Standard Error of the Mean: Note that the standard error of the mean decreases as the sample size increases Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Comparing the Population with its Sampling Distribution Sample Means Distribution n = 2 _ P(X) P(X) .3 .3 .2 .2 .1 .1 _ 18 20 22 24 A B C D X 18 19 20 21 22 23 24 X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: P(x) .25 x 18 20 22 24 A B C D Uniform Distribution Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Summary Measures of the Sampling Distribution: Developing a Sampling Distribution (continued) Summary Measures of the Sampling Distribution: Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

If sample values are not independent If the sample size n is not a small fraction of the population size N, then individual sample members are not distributed independently of one another Thus, observations are not selected independently A finite population correction is made to account for this: or The term (N – n)/(N – 1) is often called a finite population correction factor Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of is also normally distributed with and If the sample size n is not large relative to the population size N, then Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Standard Normal Distribution for the Sample Means Z-value for the sampling distribution of : where: = sample mean = population mean = standard error of the mean Z is a standardized normal random variable with mean of 0 and a variance of 1 Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (both distributions have the same mean) Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties (continued) Normal Population Distribution (i.e. is unbiased ) Normal Sampling Distribution (the distribution of has a reduced standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Sampling Distribution Properties (continued) As n increases, decreases Larger sample size Smaller sample size Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem Even if the population is not normal, …sample means from the population will be approximately normal as long as the sample size is large enough. Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem (continued) Let X1, X2, . . . , Xn be a set of n independent random variables having identical distributions with mean µ, variance σ2, and X as the mean of these random variables. As n becomes large, the central limit theorem states that the distribution of approaches the standard normal distribution Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As the sample size gets large enough… n↑ Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency Sampling Distribution (becomes normal as n increases) Variation Larger sample size Smaller sample size Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

How Large is Large Enough? For most distributions, n > 25 will give a sampling distribution that is nearly normal For normal population distributions, the sampling distribution of the mean is always normally distributed Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example Suppose a large population has mean μ = 8 and standard deviation σ = 3. Suppose a random sample of size n = 36 is selected. What is the probability that the sample mean is between 7.8 and 8.2? Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 25) … so the sampling distribution of is approximately normal … with mean = 8 …and standard deviation Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Example Solution (continued): (continued) Z X Population Distribution Sampling Distribution Standard Normal Distribution .1554 +.1554 ? ? ? ? ? ? ? ? ? ? Sample Standardize ? ? -0.4 0.4 7.8 8.2 Z X Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall Acceptance Intervals Goal: determine a range within which sample means are likely to occur, given a population mean and variance By the Central Limit Theorem, we know that the distribution of X is approximately normal if n is large enough, with mean μ and standard deviation Let zα/2 be the z-value that leaves area α/2 in the upper tail of the normal distribution (i.e., the interval - zα/2 to zα/2 encloses probability 1 – α) Then is the interval that includes X with probability 1 – α Copyright © 2013 Pearson Education, Inc. Publishing as Prentice Hall