Sampling Fundamentals 2

Sampling Process Identify Target Population Select Sampling Procedure Determine Sampling Frame Determine Sample Size

Determining Sample Size – Ad Hoc Methods Rule of thumb –Each group should have at least 100 respondents –Each sub-group should have 20 – 50 respondents Budget constraints –The question then is whether the study can be modified or cancelled Comparable studies –Find similar studies and use their sample sizes as guides

Factors determining sample size Number of groups and sub-groups in the sample that are to be analyzed Value of the study and accuracy required Cost of generating the sample Variability in the population

Revisit definitions Mean: the arithmetic average of scores on a variable –Only interval / ratio level data –Categorical data - Mode Variance: the average value of the dispersion (spread) of squared scores from the mean on a variable. Based on how a response differs from the average response Standard deviation: Square root of the variance

Basic Statistics PopulationSample Mean  X Variance  2 s 2 Standard Deviation  s Sample SizeNn 1 1   i n i n )(     i n i n s

Sample size determination In most MR problems we are interested in knowing the mean. (e.g. mean attitude scores, mean sales, etc.). We want an good estimate of the population mean Since the population mean is generally unknown, we must select the sample with care so that the sample mean will be the closest approximation to the population mean

Sample size determination We want a sample that is –Selected through random sampling –Is as large as possible

1. Normal Distribution The entire area under the curve adds up to 100%

1. Features of normal distributions 68% of responses between  + 1  95% of responses between  + 2  99.99% of responses between  + 3  Bell shaped curve Mean = Median = Mode

2. Sampling distribution of means Distribution of mean responses on an item, from every probability sample taken from the same population, the sample being taken an infinite number of times Smaller sample size = unstable means and greater variability (higher standard error of the mean) and greater sampling error Larger sample sizes = stable means and lower variability (lower standard error of the mean) and smaller sampling error Sampling distribution of means with larger sample sizes give a better approximation to the normal distribution E(X bar) = µ

3. Standard Error of the Mean Standard deviation of the sampling distribution of means  xbar =  x /  n I.e. standard error of the mean will equal the standard deviation of the population divided by the square root of the sample size –I.e. the greater the n, the smaller the standard error Therefore random sampling with a larger sample size gives a more accurate estimate

4. Sampling Error OS – TS = SE + NSE Assuming full care is taken in the research process, NSE = 0 Therefore OS – TS = SE i.e. Sampling Error is the difference between True Score and Observed Score To minimize SE – larger sample To minimize cost – smaller sample To trade off we specify SE as a percentage (e.g. 5%, 3%, 10%, etc.)

5. Interval Estimation of population mean OS – TS = SE i.e. X bar -  = SE X bar varies from sample to sample X bar + SE = interval estimate of  X bar + z  x /  n = interval estimate of  –n = sample size –z = z value at chosen confidence level

5. Size of Interval Estimate Confidence level (e.g. 90%, 95%, 99%, etc.) –The number of times the population mean must fall within the confidence interval after repeated samplings –Lower confidence levels mean smaller sample sizes and smaller intervals; Higher confidence levels mean larger sample sizes and larger intervals Population standard deviation –Generally unknown –Estimated from a previous study, a pilot, judgment or a worst case scenario

6. Statistical Sample Size Specify –S–Size of the sampling error that is desired –C–Confidence level –P–Population standard deviation X bar + SE = interval estimate of  X bar + z  x /  n = interval estimate of  SE = z  x /  n Therefore, n = Z 2  2 /(SE) 2