Sampling Methods and Sampling Distributions

Name: Sampling Methods and Sampling Distributions
Uploaded: 2017-10-08T22:08:25+00:00
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Description: Sampling Methods and Sampling Distributions

Sampling Methods and Sampling Distributions
Sampling and Sampling Distributions 4/24/2017 Sampling Methods and Sampling Distributions Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling and Sampling Distributions
4/24/2017 Learning Objectives Explain Types of Samples Describe the Properties of Estimators Explain Sampling Distribution Describe the Relationship between Populations & Sampling Distributions State the Central Limit Theorem Solve Probability Problems Involving Sampling Distributions As a result of this class, you should be able to ... Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Sampling Methods Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Types of Samples Type of Sample Non Probability Samples Selection is based on chance Subjects are chosen based on some known probabilities Eliminates or reduces bias Random refers to procedure not the data: The outcome cannot be predicted because it is dependent upon chance Non Probability Samples Do not have above characteristics Done for time and convenience Probability Probability Simple Systematic Stratified Cluster Random Judge- Quota Chunk ment Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Simple Random Sample 1. Each Population Element Has an Equal Chance of Being Selected 2. Selecting 1 Subject Does Not Affect Selecting Others 3. May Use Random Number Table, Lottery, ‘Fish Bowl’ Simple Random Use random number table Number of digits is determined by population size Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Random Number Table Columns are 01, 02 etc. (aligned vertically) Example Population size is 50. Sample size is 10. Since population size (50) has 2-digits, divide table into 2 digit numbers. Begin top left (for convenience only). 1-49, 2-28, 3-08, 4-89 (skip) 4-24, 5-35, 6-77 (skip), 6-90 (skip) 6-02, 7-83 (skip) 7-61(skip) 7-87 (skip) 7-04, 8-16, 9-57 (skip) 9-07, Population size is 100. Use 3 digit numbers. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Systematic Sample 1. Items of population arranged in some way- alphabetically, by date received 2.Every kth Element Is Selected After a Random Start within the First k Elements 3. Used in Telephone Surveys Systematic Requires all population elements Bias may occur due to periodicity In the telephone book example, unlisted numbers will not be found Example: Sampling frame is 100 individuals. You want to select 20. Select first name by random number, then every 5th person. © T/Maker Co. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Stratified Sample All Students 1. Divide Population into Subgroups Mutually Exclusive Collectively Exhaustive At Least 1 Common Characteristic of Interest Commuters Residents Stratified Assures 1. Sample reflects population in terms of criterion used for stratifying. 2. More efficient sample - sampling error is reduced. Example: College has 70% on-campus students and 30% commuters. A 100 student survey would get close to 70 on-campus students and 30 commuters. A simple random survey might get 60 on-campus and 40 commuting students. Similar to Quota sampling except that a simple random sample is drawn from each strata. Sample 2. Select Simple Random Samples from Subgroups Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Cluster Sample Companies (Clusters) Divide Population into Clusters If Managers are Elements, then Companies are Clusters Select Clusters Randomly Survey All or a Random Sample of Elements in Cluster Cluster Idea is to sample economically yet retain characteristics of probability sample. Ideally, cluster is as heterogeneous as the population. Often, characteristics of elements in cluster may be similar. Sample Mathis,Thanawala, Webster/Prentice-Hall Inc.

Nonprobability Samples
Sampling and Sampling Distributions 4/24/2017 Nonprobability Samples 1. Judgment Use Experience to Select Sample e.g., Test Markets 2. Quota Similar to Stratified Sampling Except No Random Sampling 3. Chunk (Convenience) Use Elements Most Available Judgment A fashion manufacturer selects key accounts to predict what will sell next season Quota Advantages are speed of data collection, lower costs, and convenience. Often used in laboratory experiments It is difficult to find a sample of the general population willing to visit a laboratory Chunk (Convenience) Street interviews at election time. Views represent supposedly the entire population. Need impressions of text book in an hour. Use this class to represent all students. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Errors Due to Sampling Sampling Error - occurs because sample is taken instead of census Errors are due to chance Equally likely to be too high or too low Improve by increasing sample size Nonsampling Error - Bias A directional error Can not be reduced by increasing sample size Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Errors Due to Sampling Coverage (Frame) Error Sampling Error Frame Error The sampling frame is also called the ‘working population.’ Frame error is the discrepancy between population and sampling frame. e.g., Not all students may be in phone book Sampling Error Sampling units may not perfectly represent the population. All samples vary. Sampling error is a function of sample size Systematic (Nonresponse & Measurement) Error Nonresponse, badly worded questions, interview error. Nonresponse & Measurement Error Sample Frame Total Population Planned Sample Actual (Students in (Students) (Selected Students) Sample Phone Book) Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distributions
Sampling and Sampling Distributions 4/24/2017 Sampling Distributions Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Statistical Methods Statistical Methods Descriptive Inferential Statistics Statistics Mathis,Thanawala, Webster/Prentice-Hall Inc.

Inferential Statistics
Sampling and Sampling Distributions 4/24/2017 Inferential Statistics Involves Estimation Hypothesis Testing Purpose Make Decisions about Population Characteristics Population? Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Inference Process Estimates & Tests Population Sample Statistic (`X, P ) Sample Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Estimators 1. Random Variables Used to Estimate a Population Parameter -Sample Mean, Sample Proportion, Sample Median 2. Sample Mean is an Estimator of Population Mean m If = 3 then 3 Is the Estimate of m 3. Theoretical Basis Is Sampling Distribution Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Properties of Mean Unbiasedness Mean of Sampling Distribution Equals Population Mean Efficiency Sample Mean Comes Closer to Population Mean Than Any Other Unbiased Estimator Consistency As Sample Size Increases, Variation of Sample Mean from Population Mean Decreases An estimator is a random variable used to estimate a population parameter (characteristic). Unbiasedness An estimator is unbiased if the mean of its sampling distribution is equal to the population parameter. Efficiency The efficiency of an unbiased estimator is measured by the variance of its sampling distribution. If two estimators, with the same sample size, are both unbiased, then the one with the smaller variance has greater relative efficiency. Consistency An estimator is a consistent estimator of a population parameter if the larger the sample size, the more likely it is that the estimate will come close to the parameter. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Unbiasedness P( ` X) Unbiased Biased A C ` X mx= mx mx C A Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Efficiency Sampling Distribution of Mean P( ` X) B Sampling Distribution of Median A ` X mx Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Consistency Larger Sample Size P( ` X) B Smaller Sample Size A ` X mx Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distribution
Sampling and Sampling Distributions 4/24/2017 Sampling Distribution Theoretical Probability Distribution Random Variable is Sample Statistic Sample Mean, Sample Proportion, etc. Results from Drawing All Possible Samples of a Fixed Size List of All Possible [`X, P(`X) ] Pairs Sampling Distribution of Mean Mathis,Thanawala, Webster/Prentice-Hall Inc.

Developing Sampling Distributions
Sampling and Sampling Distributions 4/24/2017 Developing Sampling Distributions Suppose There’s a Population ... Population Size, N = 4 Random Variable, X, Is # Errors in Work Values of X: 1, 2, 3, 4 Uniform Distribution Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 m X (X -m) (X - m)2 (# of errors) Population Mean and Standard Deviation Mathis,Thanawala, Webster/Prentice-Hall Inc.

Population Characteristics
Sampling and Sampling Distributions 4/24/2017 Population Characteristics Summary Measures Population Distribution N å X .3 i .2 Have students verify these numbers. m = i = 1 = 2 . 5 x .1 N .0 1 2 3 4 N ( ) å 2 X - m i x s = i = 1 = 1 . 12 x N Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Inference Process Estimates & Tests Population Sample Statistic (`X, Ps ) Sample Mathis,Thanawala, Webster/Prentice-Hall Inc.

All Possible Samples of Size n = 2
Sampling and Sampling Distributions 4/24/2017 All Possible Samples of Size n = 2 16 Samples 16 Sample Means 1st 2nd Observation 1st 2nd Observation Obs 1 2 3 4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Obs 1 2 3 4 1 1,1 1,2 1,3 1,4 2 2,1 2,2 2,3 2,4 3 3,1 3,2 3,3 3,4 4 4,1 4,2 4,3 4,4 Sample With Replacement Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distribution of All Sample Means
Sampling and Sampling Distributions 4/24/2017 Sampling Distribution of All Sample Means 1st 2nd Observation Obs 1 2 3 4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 16 Sample Means Sampling Distribution X f p(X) /16 /16 /16 /16 /16 /16 /16 Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distribution of All Sample Means
Sampling and Sampling Distributions 4/24/2017 Sampling Distribution of All Sample Means 1st 2nd Observation Obs 1 2 3 4 1.0 1.5 2.0 2.5 3.0 3.5 4.0 16 Sample Means Sampling Distribution .0 .1 .2 .3 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ` X P( X) Mathis,Thanawala, Webster/Prentice-Hall Inc.

mx N = + 1 5 4 16 2.5 . L i X å 40 4/24/2017 1.0 2.5 -1.5 2.25 1.5 -1.0 1.00 2.0 -0.5 0.25 0.0 0.00 3.0 0.5 3.5 4.0 40 10.00 X mx mx)2 (X- mx) Mathis,Thanawala, Webster/Prentice-Hall Inc.

Summary Measures of All Possible Sample Means
Sampling and Sampling Distributions 4/24/2017 Summary Measures of All Possible Sample Means m x i N X = å 1 + = 1 5 4 16 2 . L = s m x i N X = - å ( ) 2 1 79 . + 5 4 16 L 10 Have students verify these numbers. Mathis,Thanawala, Webster/Prentice-Hall Inc.

Comparison of Population & Sampling Distribution
Sampling and Sampling Distributions 4/24/2017 Comparison of Population & Sampling Distribution Population Sampling Distribution P(X) P( ` X) .3 .3 .2 .2 .1 .1 .0 .0 ` X 1 2 3 4 1 1.5 2 2.5 3 3.5 4 m = 2 . 5 m = 2 . 5 x x s = 1 . 12 s = . 79 x x Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Standard Error of Mean (Standard Deviation of the Sampling Distribution of Means) Standard Deviation of All Possible Sample Means,`X Measures Scatter in All Sample Means,`X Less Than Population Standard Deviation Formula (Sampling With Replacement) N ( ) 2 s n = å X - m s x = i x i = 1 N Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Sampling Distribution of the Sample Means Summary mx = mx Sampling is done with replacement or Population is infinite n/N < .05 Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling from Normal Populations
Sampling and Sampling Distributions 4/24/2017 Sampling from Normal Populations Population Distribution Central Tendency Dispersion Sampling With Replacement m = m x x s X = 10 s m = 50 x X s = X x n Sampling Distribution n =16 n =4 sx = 5 sx = 2.5 m - = 50 X X Mathis,Thanawala, Webster/Prentice-Hall Inc.

Standardizing Sampling Distribution of Mean
Sampling and Sampling Distributions 4/24/2017 Standardizing Sampling Distribution of Mean X - m n X x - m s x Z = = s x Sampling Distribution Standardized Normal Distribution s X s = 1 z m m = 0 ` X Z Z X Mathis,Thanawala, Webster/Prentice-Hall Inc.

Thinking Challenge 4/24/2017 You’re an operations analyst for AT&T. Long-distance telephone calls are normally distribution with mx = 8 min. & sx = 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. Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distribution Solution*
Sampling and Sampling Distributions 4/24/2017 Sampling Distribution Solution* n x s 2 25 Z X = - m 7 8 50 . Z X n x = - m s 8 2 25 50 . Sampling Distribution 8 s ` X = .4 7.8 8.2 Standardized Normal Distribution s Z = 1 .3830 .1915 -.50 .50 Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling from Normal Populations
Sampling and Sampling Distributions 4/24/2017 Sampling from Normal Populations Population Distribution Central Tendency Dispersion Sampling With Replacement m = m x x s X = 10 s m = 50 x X s = X x n Sampling Distribution n =16 sx = 2.5 n =4 sx = 5 m - = 50 X X Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling from Non-Normal Populations
Sampling and Sampling Distributions 4/24/2017 Sampling from Non-Normal Populations Population Distribution Central Tendency Dispersion Sampling With Replacement m = m x x s X = 10 s m = 50 x X s = X x n Sampling Distribution n = 4 sx= 5 n =30 sx =1.8 m - = 50 X X Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Central Limit Theorem For a population with a mean u and a standard deviation s , the sampling distribution of the means of all possible samples of size n generated from the population will be approximately normally distributed assuming that the sample size is sufficiently large. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Central Limit Theorem As sample size gets large enough (³ 30) ... sampling distribution becomes almost normal. X Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Central Limit Theorem The sampling distribution of means is a normal distribution if population is normally distributed Even if population is not normally distributed, the sampling distribution of means is approximated by a normal distribution for large n (n>30) Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Central Limit Theorem As sample size gets large enough (³ 30) ... sampling distribution becomes almost normal. X Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Proportions Categorical Variable (e.g., Gender) % Population Having a Characteristic If Two Outcomes, Binomial Distribution Possess - Don’t Possess Characteristic Sample Proportion Formula: P X n = number of successes sample siz e Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling Distribution of Proportion
Sampling and Sampling Distributions 4/24/2017 Sampling Distribution of Proportion Approximated by Normal Distribution n·p ³ 5 n·(1 - p) ³ 5 Mean Standard Error Sampling Distribution P(P ) s .3 .2 .1 .0 P m P p = .0 .2 .4 .6 .8 1.0 n where p = Population Proportion s P p = × - 1 ( ) Mathis,Thanawala, Webster/Prentice-Hall Inc.

Standardizing Sampling Distribution of Proportion
Sampling and Sampling Distributions 4/24/2017 Standardizing Sampling Distribution of Proportion P - m p P - P Z @ = s p × ( 1 - p ) P n Sampling Distribution Standardized Normal Distribution s s = 1 P z m m = 0 P Z P Z Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Thinking Challenge You’re manager of a bank. 40% of depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ? © T/Maker Co. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Solution* P(.40 £ P £ .43) ü n·p ³ 5 n·(1 - p) ³ 5 Z P p n @ - × = ( ) . 1 43 40 200 87 Sampling Distribution Z m = 0 s = 1 Standardized Normal Distribution s = .0346 P .3078 P m = .40 .43 .87 P Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sampling from Finite Populations
Sampling and Sampling Distributions 4/24/2017 Sampling from Finite Populations Modify Standard Error if Sample Size (n) Is Large Relative to Population Size (N) n > .05·N (or n/N > .05) Use Finite Population Correction (fpc) Factor for Standard Errors if n/N > .05 × s P p n = - 1 ( ) s x n = × N n - 1 N n - 1 ) ( Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Sampling Distribution of the Sample Means Summary mx = mx Sampling is done with replacement or Population is infinite n/N < .05 Sampling is without replaacement and Population is finite n/N > .05 Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Thinking Challenge You’re manager of a bank. 40% of all 1000 depositors have multiple accounts. You select a random sample of 200 customers. What is the probability that the sample proportion of depositors with multiple accounts would be between 40% & 43% ? © T/Maker Co. Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Solution* P(.40 £ P £ .43) P - p . 43 - . 40 Z @ = = . 97 p × ( 1 - p ) N - n . 40 × ( 1 - . 40 ) 1000 - 200 × n N - 1 200 1000 - 1 Sampling Distribution Compare these results to previous problem with a very large population. Standardized Distribution s = .0310 s = 1 P Z .3340 P m = 0 .97 Z m = .40 .43 P Z Mathis,Thanawala, Webster/Prentice-Hall Inc.

Selecting a Sample Size
Sampling and Sampling Distributions 4/24/2017 Selecting a Sample Size Mathis,Thanawala, Webster/Prentice-Hall Inc.

Selecting a Sample Size
Sampling and Sampling Distributions 4/24/2017 Selecting a Sample Size The Degree of Cofidence Selected The Maximum Allowable Error The Population Standard Deviation Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Sample Size for Means E is the allowable error z is the z score associated with degree of confidence s is the population standard deviation Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 The marketing manager would like to estimate the population mean annual usage of home heating oil to within 50 gallons of the true value and desires to be 95% confident of correctly estimating the true mean. Based on a previous study taken last year,the marketing manager feels that the standard deviation can be estimated as 325 gallons. What is the sample size need to obtain these results? Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 z = 1.96 Confidence = 95% E = 50 s = 325 n= 163 homes need to be sampled Mathis,Thanawala, Webster/Prentice-Hall Inc.

Sample Size for Proportions
Sampling and Sampling Distributions 4/24/2017 Sample Size for Proportions E is the maximum allowable error z is the z value associated with the degree of confidence p is the estimated proportion Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 A political pollister would like to estimate the proportion of voters who will vote for the Democratic candidate in a presidential campaign. The pollster would like 95% confidence that her prediction is correct to within of the true proportion. What sample size is needed? Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Confidence = 95% E = .04 p = unknown use p = .5 n = 601 voters Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 Conclusion Examined Sampling Methods Described the Properties of Estimators Explained Sampling Distribution Described the Relationship between Populations & Sampling Distributions Stated the Central Limit Theorem Solved Probability Problems Involving Sampling Distributions Mathis,Thanawala, Webster/Prentice-Hall Inc.

4/24/2017 End of Chapter Any blank slides that follow are blank intentionally. Mathis,Thanawala, Webster/Prentice-Hall Inc.

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