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Basic Business Statistics Chapter 7 Sampling and Sampling Distributions Assoc. Prof. Dr. Mustafa Yüzükırmızı Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 6-1 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, Assoc. Prof. Mustafa Yuzukirmizi 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-3 A Sampling Process Begins With A Sampling Frame The sampling frame is a listing of items that make up the population Frames are data sources such as population lists, directories, or maps 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-4 Types of Samples Samples Non-Probability Samples Judgment Convenience Probability Samples Simple Random Stratified Systematic Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Cluster 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. In a judgment sample, you get the opinions of preselected experts in the subject matter. Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 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 Random Systematic Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Stratified Cluster 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-8 Selecting a Simple Random Sample Using A Random Number Table Sampling Frame For Population With 850 Items Item Name Item # Bev R. Ulan X. . . . . Joann P. Paul F. 001 002 . . . . 849 850 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Portion Of A Random Number Table 49280 88924 35779 00283 81163 07275 11100 02340 12860 74697 96644 89439 09893 23997 20048 49420 88872 08401 The First 5 Items in a simple random sample Item # 492 Item # 808 Item # 892 -- does not exist so ignore Item # 435 Item # 779 Item # 002 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-10 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 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, Assoc. Prof. Mustafa Yuzukirmizi 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 All items in the selected clusters can be used, or items can be chosen from a cluster using another probability sampling technique A common application of cluster sampling involves election exit polls, where certain election districts are selected and sampled. Population divided into 16 clusters. Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Randomly selected clusters for sample 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 Less efficient (need larger sample to acquire the same level of precision) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-15 Types of Survey Errors Coverage error or selection bias Non response error or bias People who do not respond may be different from those who do respond Sampling error Exists if some groups are excluded from the frame and have no chance of being selected 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”) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-16 Types of Survey Errors (continued) Coverage error Excluded from frame Non response error Follow up on nonresponses Sampling error Random differences from sample to sample Measurement error Bad or leading question Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-18 Developing a Sampling Distribution Assume there is a population … Population size N=4 Random variable, X, is age of individuals Values of X: 18, 20, 22, 24 (years) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi A B C D Chap 7-19 Developing a Sampling Distribution (continued) Summary Measures for the Population Distribution: X μ P(x) i N .3 18 20 22 24 21 4 σ (X μ) i N .2 .1 0 2 2.236 18 20 22 24 A B C D x Uniform Distribution Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-20 Developing a Sampling Distribution (continued) Now consider all possible samples of size n=2 1st Obs 16 Sample Means 2nd Observation 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 1st 2nd Observation Obs 18 20 22 24 22 22,18 22,20 22,22 22,24 18 18 19 20 21 24 24,18 24,20 24,22 24,24 20 19 20 21 22 16 possible samples (sampling with replacement) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 22 20 21 22 23 24 21 22 23 24 Chap 7-21 Developing a Sampling Distribution (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 _ P(X) .3 .2 .1 0 18 19 20 21 22 23 24 _ X (no longer uniform) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-22 Developing a Sampling Distribution (continued) Summary Measures of this Sampling Distribution: μX X i 18 19 19 24 21 σX N 16 2 ( X μ ) i X N (18 - 21)2 (19 - 21)2 (24 - 21)2 1.58 16 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-23 Comparing the Population Distribution to the Sample Means Distribution Population N=4 μ 21 σ 2.236 Sample Means Distribution n=2 μX 21 σ X 1.58 _ P(X) .3 P(X) .3 .2 .2 .1 .1 0 18 20 22 24 A B C D Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi X 0 18 19 20 21 22 23 24 _ X 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-25 Sample Mean Sampling Distribution: If the Population is Normal If a population is normal with mean μ and standard deviation σ, the sampling distribution of X is also normally distributed with μX μ Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi and σ σX n Chap 7-26 Z-value for Sampling Distribution of the Mean Z-value for the sampling distribution of X : Z where: ( X μX ) σX ( X μ) σ n X = sample mean μ = population mean σ = population standard deviation n = sample size Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-27 Sampling Distribution Properties μx μ (i.e. x is unbiased ) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Normal Population Distribution μ x μx x Normal Sampling Distribution (has the same mean) Chap 7-28 Sampling Distribution Properties (continued) As n increases, Larger sample size σ x decreases Smaller sample size μ Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi x Chap 7-29 Determining An Interval Including A Fixed Proportion of the Sample Means Find a symmetrically distributed interval around µ that will include 95% of the sample means when µ = 368, σ = 15, and n = 25. Since the interval contains 95% of the sample means 5% of the sample means will be outside the interval Since the interval is symmetric 2.5% will be above the upper limit and 2.5% will be below the lower limit. From the standardized normal table, the Z score with 2.5% (0.0250) below it is -1.96 and the Z score with 2.5% (0.0250) above it is 1.96. Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-30 Determining An Interval Including A Fixed Proportion of the Sample Means (continued) Calculating the lower limit of the interval σ 15 XL μ Z 368 (1.96) 362.12 n 25 Calculating the upper limit of the interval σ 15 XU μ Z 368 (1.96) 373.88 n 25 95% of all sample means of sample size 25 are between 362.12 and 373.88 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-31 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: μx μ Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi and σ σx n Chap 7-32 Central Limit Theorem As the sample size gets large enough… n↑ the sampling distribution becomes almost normal regardless of shape of population x Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-33 Sample Mean Sampling Distribution: If the Population is not Normal (continued) Population Distribution Sampling distribution properties: Central Tendency μx μ σ σx n Variation Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi μ x Sampling Distribution (becomes normal as n increases) Larger sample size Smaller sample size μx x Chap 7-34 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 For normal population distributions, the sampling distribution of the mean is always normally distributed Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-35 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, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-36 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 approximately normal x is … with mean μx = 8 σ 3 …and standard deviation σ x n 36 0.5 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-37 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 Distribution ??? ? ?? ? ? ? ? ? μ8 Sampling Distribution Standard Normal Distribution Sample .1554 +.1554 Standardize ? X Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi 7.8 μX 8 8.2 x -0.4 μz 0 0.4 Z Chap 7-38 Population Proportions π = the proportion of the population having some characteristic p Sample proportion ( p ) provides an estimate of π: X number of items in the sample having the characteri stic of interest n sample size 0≤ p≤1 p is approximately distributed as a normal distribution when n is large (assuming sampling with replacement from a finite population or without replacement from an infinite population) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-39 Sampling Distribution of p Approximated by a normal distribution if: and 0 n(1 π ) 5 μp π Sampling Distribution .3 .2 .1 0 nπ 5 where P( ps) and .2 .4 .6 8 1 p π(1 π ) σp n (where π = population proportion) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-40 Z-Value for Proportions Standardize p to a Z value with the formula: p Z σp Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi p (1 ) n Chap 7-41 Example If the true proportion of voters who support Proposition A is π = 0.4, what is the probability that a sample of size 200 yields a sample proportion between 0.40 and 0.45? i.e.: if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-42 Example (continued) if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Find σ p : σ p (1 ) n 0.4(1 0.4) 0.03464 200 0.45 0.40 0.40 0.40 Convert to P(0.40 p 0.45) P Z standardized 0.03464 0.03464 normal: P(0 Z 1.44) Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-43 Example (continued) if π = 0.4 and n = 200, what is P(0.40 ≤ p ≤ 0.45) ? Use standardized normal table: P(0 ≤ Z ≤ 1.44) = 0.4251 Standardized Normal Distribution Sampling Distribution 0.4251 Standardize 0.40 0.45 Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi p 0 1.44 Z Chap 7-44 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 Described the sampling distribution of a proportion Calculated probabilities using sampling distributions Basic Business Statistics, Assoc. Prof. Mustafa Yuzukirmizi Chap 7-45