Business Statistics: A First Course Chapter 7 PDF

Business Statistics: A First Course 5th Edition Chapter 7 Sampling and Sampling Distributions Basic Business Statisti...

Business Statistics: A First Course 5th Edition Chapter 7 Sampling and Sampling Distributions Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc. Chap 7-1  Learning Objectives In this chapter, you learn: To distinguish between different sampling methods The concept of the sampling distribution To compute probabilities related to the sample mean and the sample proportion The importance of the Central Limit Theorem Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-2  Why Sample? Selecting a sample is less time-consuming than selecting every item in the population (census). Selecting a sample is less costly than selecting every item in the population. An analysis of a sample is less cumbersome and more practical than an analysis of the entire population. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-3  A Sampling Process Begins With A Sampling Frame The sampling frame is a listing of items that make up the population Inaccurate or biased results can result if a frame excludes certain portions of the population Using different frames to generate data can lead to dissimilar conclusions Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-4  Types of Samples Samples Non-Probability Probability Samples Samples Simple Stratified Judgment Convenience Random Systematic Cluster Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-5  Types of Samples: Nonprobability Sample In a nonprobability sample, items included are chosen without regard to their probability of occurrence. In convenience sampling, items are selected based only on the fact that they are easy, inexpensive, or convenient to sample.(tires in warehouse) In a judgment sample, you get the opinions of pre- selected experts in the subject matter, you can not generalize their results to general public. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-6  Types of Samples: Probability Sample In a probability sample, items in the sample are chosen on the basis of known probabilities. Probability Samples Simple Systematic Stratified Cluster Random Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-7  Probability Sample: Simple Random Sample Every individual or item from the frame has an equal chance of being selected Selection may be with replacement (selected individual is returned to frame for possible reselection) or without replacement (selected individual isn’t returned to the frame). Samples obtained from table of random numbers or computer random number generators. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-8  Selecting a Simple Random Sample Using A Random Number Table Portion Of A Random Number Table Sampling Frame For 49280 88924 35779 00283 81163 07275 Population With 850 11100 02340 12860 74697 96644 89439 Items 09893 23997 20048 49420 88872 08401 Item Name Item # The First 5 Items in a simple Bev R. 001 random sample Ulan X. 002 Item # 492.. Item # 808.. Item # 892 -- does not exist so ignore.. Item # 435.. Item # 779 Joann P. 849 Item # 002 Paul F. 850 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-9  Probability Sample: Systematic Sample Decide on sample size: n Divide frame of N individuals into groups of k individuals: k=N/n Randomly select one individual from the 1st group Select every kth individual thereafter N = 40 First Group n=4 k = 10 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-10 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-11  Probability Sample: Stratified Sample Divide population into two or more subgroups (called strata) according to some common characteristic A simple random sample is selected from each subgroup, with sample sizes proportional to strata sizes Samples from subgroups are combined into one This is a common technique when sampling population of voters, stratifying across racial or socio-economic lines. Population Divided into 4 strata Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-12  Probability Sample Cluster Sample Population is divided into several “clusters,” each representative of the population A simple random sample of clusters is selected A common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled. Population divided into 16 clusters. Randomly selected clusters for sample Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-13  Probability Sample: Comparing Sampling Methods Simple random sample and Systematic sample Simple to use May not be a good representation of the population’s underlying characteristics Stratified sample Ensures representation of individuals across the entire population Cluster sample More cost effective than simple random sampling, especially the population is spread over a wide geographic region Less efficient (need larger sample to acquire the same level of precision) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-14  Evaluating Survey Worthiness What is the purpose of the survey? Is the survey based on a probability sample? Coverage error – appropriate frame? Nonresponse error – follow up Measurement error – good questions elicit good responses Sampling error – always exists Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-15  Types of Survey Errors Coverage error or selection bias Exists if some groups are excluded from the frame and have no chance of being selected Non response error or bias People who do not respond may be different from those who do respond Sampling error Variation from sample to sample will always exist Measurement error Due to weaknesses in question design, respondent error, and interviewer’s effects on the respondent (“Hawthorne effect”- respondent feels obligated to please the interviewer) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-16  Types of Survey Errors (continued) Excluded from Coverage error frame Follow up on Non response error nonresponses Random Sampling error differences from sample to sample Measurement error Bad or leading question Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-17  Sampling Distributions A sampling distribution is a distribution of all of the possible values of a sample statistic for a given size sample selected from a population. For example, suppose you sample 50 students from your college regarding their mean GPA. If you obtained many different samples of 50, you will compute a different mean for each sample. We are interested in the distribution of all potential mean GPA we might calculate for any given sample of 50 students. Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-18  Developing a Sampling Distribution Assume there is a population … A C D Population size N=4 B Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-19  Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: μ  X i P(x) N.3 18  20  22  24.2  21 4.1 0 σ  (X  μ) i 2 2.236 18 20 22 24 x N A B C D Uniform Distribution Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-20  Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 16 Sample 1st 2nd Observation Obs Means 18 20 22 24 1st 2nd Observation 18 18,18 18,20 18,22 18,24 Obs 18 20 22 24 20 20,18 20,20 20,22 20,24 18 18 19 20 21 22 22,18 22,20 22,22 22,24 20 19 20 21 22 24 24,18 24,20 24,22 24,24 16 possible samples 22 20 21 22 23 (sampling with 24 21 22 23 24 replacement) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-21  Developing a Sampling Distribution (continued) Sampling Distribution of All Sample Means 16 Sample Means Sample Means Distribution 1st 2nd Observation _ Obs 18 20 22 24 P(X).3 18 18 19 20 21.2 20 19 20 21 22.1 22 20 21 22 23 0 _ 24 21 22 23 24 18 19 20 21 22 23 24 X (no longer uniform) Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-22  Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: μX   X i 18  19  19    24  21 N 16 σX   ( X i  μ X ) 2 N (18 - 21)2  (19 - 21)2    (24 - 21)2  1.58 16 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-23  Comparing the Population Distribution to the Sample Means Distribution Population Sample Means Distribution N=4 n=2 μ 21 σ 2.236 μX 21 σ X 1.58 _ P(X) P(X).3.3.2.2.1.1 0 X 0 18 19 20 21 22 23 24 _ 18 20 22 24 X A B C D Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-24  Sample Mean Sampling Distribution: 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: (This assumes that sampling is with replacement or sampling is without replacement from an infinite population) σ σX  n Note that the standard error of the mean decreases as the sample size increases Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-25  Sample Mean Sampling Distribution: If the Population is Normal If a population is normally distributed with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with σ μ X μ σX  and n Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-26  Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of X: ( X  μX ) ( X  μ) Z  σX σ n where: X = sample mean μ = population mean σ = population standard deviation n = sample size Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-27  Sampling Distribution Properties Normal Population μx μ Distribution μ x (i.e. x is unbiased ) Normal Sampling Distribution (has the same mean) μx x Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-28  Sampling Distribution Properties (continued) As n increases, Larger σ x decreases sample size Smaller sample size μ x Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-29  Sample Mean Sampling Distribution: If the Population is not Normal We can apply the 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. Properties of the sampling distribution: σ μ x μ and σx  n Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-30  Central Limit Theorem the sampling As the n↑ distribution sample becomes size gets almost normal large regardless of enough… shape of population x Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-31  Sample Mean Sampling Distribution: If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency μ x μ μ x Sampling Distribution Variation σ (becomes normal as n increases) σx  Larger n Smaller sample size sample size μx x Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-32  How Large is Large Enough? For most distributions, n > 30 will give a sampling distribution that is nearly normal For fairly symmetric distributions, n > 15 will usually give a sampling distribution is almost normal For normal population distributions, the sampling distribution of the mean is always normally distributed Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-33  Example Suppose a 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? Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-34  Example (continued) Solution: Even if the population is not normally distributed, the central limit theorem can be used (n > 30) … so the sampling distribution of x is approximately normal … with mean μx = 8 σ 3 …and standard deviation σ x   0.5 n 36 Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-35  Example (continued) Solution (continued):    7.8 - 8 X -μ 8.2 - 8  P(7.8  X  8.2)  P     3 σ 3   36 n 36   P(-0.4  Z  0.4)  0.3108 Population Sampling Standard Normal Distribution Distribution Distribution.1554 ??? +.1554 ? ?? ? ? ? ? ? Sample Standardize ? -0.4 0.4 μ 8 X 7.8 μX 8 8.2 x μz 0 Z Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-36  Chapter Summary Discussed probability and nonprobability samples Described four common probability samples Examined survey worthiness and types of survey errors Introduced sampling distributions Described the sampling distribution of the mean For normal populations Using the Central Limit Theorem Calculated probabilities using sampling distributions Basic Business Statistics, 11e © 2009 Prentice-Hall, Inc.. Chap 7-37