7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.1 Sampling for Estimation Instructor: Ron S. Kenett Course Website: Course textbook: MODERN INDUSTRIAL STATISTICS, Kenett and Zacks, Duxbury Press, 1998

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.2 Course Syllabus Understanding Variability Variability in Several Dimensions Basic Models of Probability Sampling for Estimation of Population Quantities Parametric Statistical Inference Computer Intensive Techniques Multiple Linear Regression Statistical Process Control Design of Experiments

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.3 Error Sampling Nonsampling Standard error of the mean of the proportion Standardized individual value sample mean Finite Population Correction (FPC) Probability sample Simple random sample Systematic sample Stratified sample Cluster sample Nonprobability sample Convenience sample Quota sample Purposive sample Judgment sample Key Terms

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.4 Key Terms Unbiased estimator Point estimates Interval estimates Interval limits Confidence coefficient Confidence level Accuracy Degrees of freedom (df) Maximum likely sampling error

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.5 Types of Samples Simple random Systematic Every person has an equal chance of being selected. Best when roster of the population exists. Randomly enter a stream of elements and sample every kth element. Best when elements are randomly ordered, no cyclic variation. Probability, or Scientific, Samples: Each element to be sampled has a known (or calculable) chance of being selected.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.6 Types of Samples Stratified Cluster Randomly sample elements from every layer, or stratum, of the population. Best when elements within strata are homogeneous. Randomly sample elements within some of the strata. Best when elements within strata are heterogeneous. Probability, or Scientific, Samples: Each element to be sampled has a known (or calculable) chance of being selected.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.7 Types of Samples Convenience Quota Elements are sampled because of ease and availability. Elements are sampled, but not randomly, from every layer, or stratum, of the population. Nonprobability Samples: Not every element has a chance to be sampled. Selection process usually involves subjectivity.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.8 Types of Samples Purposive Judgment Elements are sampled because they are atypical, not representative of the population. Elements are sampled because the researcher believes the members are representative of the population. Nonprobability Samples: Not every element has a chance to be sampled. Selection process usually involves subjectivity.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.9 Distribution of the Mean When the population is normally distributed Shape: Regardless of sample size, the distribution of sample means will be normally distributed. Center: The mean of the distribution of sample means is the mean of the population. Sample size does not affect the center of the distribution. Spread: The standard deviation of the distribution of sample means, or the standard error, is. n x  

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.10 The Standardized Mean The standardized z-score is how far above or below the sample mean is compared to the population mean in units of standard error. “ How far above or below ” sample mean minus µ “ In units of standard error ” divide by Standardized sample mean n x z      – error standard mean sample n 

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.11 Distribution of the Mean When the population is not normally distributed Shape: When the sample size taken from such a population is sufficiently large, the distribution of its sample means will be approximately normally distributed regardless of the shape of the underlying population those samples are taken from. According to the Central Limit Theorem, the larger the sample size, the more normal the distribution of sample means becomes.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.12 Distribution of the Mean When the population is not normally distributed Center: The mean of the distribution of sample means is the mean of the population, µ. Sample size does not affect the center of the distribution. Spread: The standard deviation of the distribution of sample means, or the standard error, is. n x  

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.13 Distribution of the Proportion When the sample statistic is generated by a count not a measurement, the proportion of successes in a sample of n trials is p, where Shape: Whenever both n  and n(1 –  ) are greater than or equal to 5, the distribution of sample proportions will be approximately normally distributed.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.14 Distribution of the Proportion When the sample proportion of successes in a sample of n trials is p, Center: The center of the distribution of sample proportions is the center of the population, . Spread: The standard deviation of the distribution of sample proportions, or the standard error, is  p   ׳ (1–  ) n.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.15 Distribution of the Proportion The standardized z-score is how far above or below the sample proportion is compared to the population proportion in units of standard error. “ How far above or below ” sample p –  “ In units of standard error ” divide by Standardized sample proportion n p z )–1( – error standard proportion sample  ׳      n p )–1(  ׳  

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.16 Finite Population Correction Finite Population Correction (FPC) Factor: Rule of Thumb: Use FPC when n > 5% N. Apply to: Standard errors of mean and proportion.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.17 Unbiased Point Estimates PopulationSample ParameterStatistic Formula Mean, µ Variance,   Proportion,  xx  x i n 1– 2 )–( 22 n x i x ss  pp  x successes n trials

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.18 Confidence Intervals: Confidence Intervals: µ,  Known where = sample mean ASSUMPTION:  = population standard infinite population deviation n = sample size z = standard normal score for area in tail =  /2 n zxx n zxx zzz  ׳  ׳  –: 0–:

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.19 Confidence Intervals: Confidence Intervals: µ,  Unknown where = sample mean ASSUMPTION: s = sample standard Population deviation approximately n = sample size normal and t = t-score for area infinite in tail =  /2 df = n – 1 n s txx n s txx ttt ׳  ׳  –: 0–:

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.20 Confidence Intervals on Confidence Intervals on  where p = sample proportion ASSUMPTION: n = sample size n p  5, z = standard normal score n (1 – p)   5, for area in tail =  /2and population infinite nn pp zpp pp zpp )–1()–1( –: ׳  ׳

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.21 Confidence Intervals for Finite Populations Mean: or Proportion:

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.22 Interpretation of Confidence Intervals Repeated samples of size n taken from the same population will generate (1 –  )% of the time a sample statistic that falls within the stated confidence interval. OR We can be (1 –  )% confident that the population parameter falls within the stated confidence interval.

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.23 Sample Size Determination for Infinite Populations Mean: Note  is known and e, the bound within which you want to estimate µ, is given. The interval half-width is e, also called the maximum likely error: Solving for n, we find: 2 22 e z n n ze   ׳  ׳ 

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.24 Sample Size Determination for Finite Populations Mean: Note  is known and e, the bound within which you want to estimate µ, is given. where n = required sample size N = population size z = z-score for (1 –  )% confidence

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.25 Sample Size Determination of for Infinite Populations Sample Size Determination of  for Infinite Populations Proportion: Note e, the bound within which you want to estimate , is given. The interval half-width is e, also called the maximum likely error: Solving for n, we find: 2 )–1( 2 )–1( e ppz n n pp ze  ׳ 

7/2/2015 (c) 2001, Ron S. Kenett, Ph.D.26 Sample Size Determination of for Sample Size Determination of  for Finite Populations Mean: Note e, the bound within which you want to estimate , is given. where n = required sample size N = population size z = z-score for (1 –  )% confidence p = sample estimator of  n  p(1–p) e 2 z 2  p(1–p) N