Basic Sampling & Review of Statistics

Basic Sampling What is a sample?  Selection of a subset of elements from a larger group of objects Why use a sample?  Saves Time Money  Accuracy Lessens non-sampling error

Basic Sampling Major definitions  Sample population – entire group of people from whom the researcher needs to obtain information  Sample element -- unit from which information is sought (consumers)  Sampling unit -- elements available for selection during the sampling process (consumers who are in the US at the time of the study)  Sampling frame -- list of all sampling units available for selection to the sample (list of all consumers who are in the US at the time of the study)  Sampling error -- difference between population response and sample response  Non-sampling error – all other errors that emerge during data collection

Basic Sampling Procedure for selecting a sample  Define the population – who (or what) we want data from  Identify the sampling frame – those available to get data from  Select a sampling procedure – how we are going to obtain the sample  Determine the sample size (n)  Draw the sample  Collect the data

Basic Sampling General Types of Samples  Non-probability – selection of element to be included in final sample is based on judgment of the researcher  Probability – each element of population has a known chance of being selected Selection of element is chosen on the basis of probability  Characteristics of probability samples Calculation of sampling error (+ or - z (  x )) Make inferences to the population as a whole

Non-Probability samples Convenience  Sample is defined on the basis of the convenience of the researcher Judgment  Hand-picked sample because elements are thought to be able to provide special insight to the problem at hand Snowball  Respondents are selected on the basis of referrals from other sample elements Often used in more qualitative/ethnographic type studies Quota  Sample chosen such that a specified proportion of elements possessing certain characteristics are approximately the same as the proportion of elements in the universe

Probability Samples Simple random sample (SRS)  Assign a number to each sampling unit  Use random number table Systematic Sample  Easy alternative to SRS Stratified sample  Divide population into mutually exclusive strata  Take a SRS from each strata

Probability Samples Cluster sample  Divide population into mutually exclusive clusters Select a SRS of clusters One-stage -- measure all members in the cluster Two-stage --measure a SRS within the cluster Area sample  One-stage -- Choose an SRS of blocks in an area; sample everyone on the block  Two-stage -- Choose an SRS of blocks in an area; select an SRS of houses on the block

Random Number Table

Hypothetical Sample Populations Responden t Number Income ($,000) Education (Years) Yogurt Consumptio n (Cartons/Yea r) Satisfaction Level (1 – 7) City Madison Milwaukee Milwaukee Milwaukee Madison Milwaukee Madison Madison Madison Milwaukee Other Madison Milwaukee Milwaukee Madison Other Milwaukee Milwaukee Madison Madison

Review of Statistics Probability Samples – note that statistical error can be computed when they are used  Thus, need to know about statistics Descriptive statistics  Estimates of descriptions of a population Statistical terms used in sampling  Mean (  or x  x i /n  Variance (  2 or s 2 ) --  x i -x) 2 /n - 1  Standard Deviation (  or s) – Square Root (Variance)

Review of Statistics Inferential Statistics  Terms Parameter --  Statistic -- x  Sample Statistics Best estimate of population parameter Why? -- Central Limit Theorem

Review of Statistics Central Limit Theorem  Based on the distribution of the means of numerous samples Sampling Distribution of Means  Theorem states: as sample size (n) approaches infinity (gets large), the sampling distribution of means becomes normally distributed with mean (  ) and standard deviation (  √  n) Allows the calculation of sampling error ( s  √  n)  Thus a confidence interval can be calculated

Review of Statistics Confidence interval -- tells us how close, based on n and the sampling procedure, how close the sampling mean (x) is to the population mean (  )  Formula: x - z (  x ) < (  ) < x + z (  x )  z-values: 90% % %

Review of Statistics Confidence interval -- interpretation  For the same sampling procedure, 95 out 100 calculated confidence intervals would include the true mean (  )

Sample Size Sample size and total error  Larger n increases probability of non-sampling error  Larger n reduces sampling error (  √  n)  Effect on n on total error?  Can pre-determine the level of error (by setting n) Depends mainly on the method of analysis

Sample Size Sample size when research objective is estimate a population parameter  CI = x ± z S x  CI = x ± 1.96 (s/ √n)  n = x ± z 2 s 2 / h 2  n = (1.96) 2 s 2 / h 2  n = (3.84) s 2 / h 2 s = expected standard deviation h = absolute precision of the estimate (or with of the desired confidence interval)

Sample Size (Sample Exercise) n = (1.96) 2 s 2 / h 2  S = 7.5  h =.50 n = (3.84) (56.25)/.025 n = 216/.025 n = 8640 What if s = 10; h = 1  n = (3.84) (100)/1  n = 384

Sample Size (Conclusion) Unaffected by size of universe Affected by  Choice of Desired Precision of Confidence Interval  Estimate of standard deviation

Sample Size Sample size estimation  With cross-tabulation based research  Objective is to get a minimum of 25 subjects per cell  Must estimate relationship up front – what is smallest cell <3030+Total Fem Male.30Small est (.10).40 Total.55.45

Sample Size Know smallest cell size should be 25 Calculate Total Sample size 25 is 10% of sample Total Sample size  25 =.10 n  25/.10 = n  250 = n <3030+Total Fem Male25 Total