Sample Size Determination

Learning Objectives To learn the financial and statistical issues in the determination of sample size To discover methods for determining sample size To gain an appreciation of a normal distribution John Wiley & Son, Inc

Learning Objectives To understand population, sample, and sampling distributions. To distinguish between point and interval estimates To recognize problems involving sampling means and proportions. John Wiley & Son, Inc

Determining Sample Size for Probability Samples The financial and statistical issues in the determination of sample size Financial, Statistical, and Managerial Issues As a general rule: The larger the sample, the smaller the sampling error. Larger samples cost more on a liner basis; however, the sampling error decreases at a rate equal to the square root of the relative increase in sample size. Before trying to determine the size of the sample, the confidence intervals need to be decided. John Wiley & Son, Inc

Determining Sample Size for Probability Samples The financial and statistical issues in the determination of sample size Budget Available Sample Size—a project is often determined by the available budget Alternative Data Collection Approaches budget constraints force the researcher to explore and consider the value of information in relation to its cost Rules of Thumb Desired sampling error - Similar Studies Past experience - A gut feeling John Wiley & Son, Inc

Determining Sample Size for Probability Samples The financial and statistical issues in the determination of sample size Number of Subgroups To Be Analyzed Subgroups Samples size Minimum needs Traditional Statistical Methods An estimate of the population standard deviation. The acceptable level of sampling error. The desired level of confidence that the sample will fall within a certain range of the true population values. John Wiley & Son, Inc

The Normal Distribution To gain an appreciation of a normal distribution The Normal Distribution General Properties for the Normal Distribution Crucial to Classical Statistical Inference Reasons For Its Importance Many variables have probability distributions that are close to the normal distribution Central Limit Theorem—distribution of a large number of sample means or sample proportions will approximate a normal distribution, regardless of the distribution of the population from which they were drawn John Wiley & Son, Inc

The Normal Distribution To gain an appreciation of a normal distribution The Normal Distribution Important Characteristics of the Normal Distribution The normal distribution is bell-shaped and has only one mode. Symmetrical about the mean Uniquely defined by its mean and standard deviation. The total area is equal to one. The area between any two values of a variable equals the probability of observing a value in that range when randomly selecting an observation from the distribution. The area between the mean and a given number of standard deviations from the mean is the same for all normal distributions John Wiley & Son, Inc

The Normal Distribution To gain an appreciation of a normal distribution The Normal Distribution The Standard Normal Distribution The same features as any normal distribution. The mean is equal to zero The standard deviation is equal to one. John Wiley & Son, Inc

X = value of the variable  = mean of the variable To gain an appreciation of a normal distribution The Normal Distribution value of the variable - mean of the variable Z= standard deviation of the variable X -  Z =  where X = value of the variable  = mean of the variable  = standard deviation of the variable John Wiley & Son, Inc

To understand population, sample, and sampling distributions. Population and Sample Distributions To understand population, sample, and sampling distributions. Purpose of Conducting a Survey To make inferences Sample Population Distribution A frequency distribution of all the elements of a population. It has a mean (µ) Sample Distribution A frequency distribution of all the elements of an individual sample. The mean x The standard divergence—S John Wiley & Son, Inc

Sampling Distributions To understand population, sample, Of The Mean To understand population, sample, and sampling distributions. If the samples are sufficiently large and random, the resulting distribution of sample means will approximate a normal distribution. The distribution of the means of a large number of random samples taken from virtually any population approaches a normal distribution with a mean equal to  and a standard deviation equal to: s x =  n √ John Wiley & Son, Inc

Sampling Distributions To understand population, sample, Of The Mean To understand population, sample, and sampling distributions. Basic Concepts Characteristics of a Large Simple Random Sample 30 or more observations Distribution is a normal distribution Distribution has a mean equal to the population mean Distribution has a standard deviation, referred to as the standard error of the mean, equal to the population standard deviation divided by the square root of the sample =  n √  x John Wiley & Son, Inc

Sampling Distributions To understand population, sample, Of The Mean To understand population, sample, and sampling distributions. Making Inferences on the Basis of a Single Sample A 68.26% probability that a random sample will produce an estimate of the population mean within one standard error (±) of the true mean 95.44 % probability that a random sample will produce an estimate of the population mean within two standard error (±) of the true mean 99.74 % probability that a random sample will produce an estimate of the population mean within three standard error (±) of the true mean John Wiley & Son, Inc

Sampling Distributions To distinguish between point Of The Mean To distinguish between point and interval estimates. Point Estimates Inferences regarding the sampling error associated with a particular estimate of the population value. Sampling Error The distance between the x and Χ Interval Estimate The likelihood that a population value falls within a certain range. Confidence Interval The interval that in all probability, includes the true population value John Wiley & Son, Inc

Sampling Distributions To distinguish between point Of The Mean To distinguish between point and interval estimates. Method of Deriving Interval Estimates Draw a random sample from the population interest and calculating the mean of that sample There is a 68.26% probability that the sample mean lies within one σ x Below is the statement symbolically 1 x x <  + John Wiley & Son, Inc

Sampling Distributions Of The Mean To recognize problems involving sampling means and proportions. Interest in Estimating Proportions or Percentages Rather than or in addition to estimating means. Examples The percentage of the population that is aware of a particular The percentage of the population that accesses the Internet one or more times in an average week The percentage of the population that has visited a fast-food restaurant four or more times in the past 30 days John Wiley & Son, Inc

Sampling Distributions Of The Mean To recognize problems involving sampling means and proportions. Sampling Distribution of the proportion A relative frequency distribution of the sample proportions of a large number of random samples of a given size drawn from a particular population. Approximates a normal distribution The mean proportion is equal to the population proportion. Standard error computed as: P (1-P) Sp = n √ John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Problems Involving Means The formula for calculating the required sample size for problems that involve the estimation of a mean: n = Z2 2 E2 where: Z = level of confidence expressed in standard errors  = population standard deviation E = acceptable amount of sampling error John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Information needed to compute the sample size Allowable sampling error Population standard deviation John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Four Methods Used to Deal with the Estimate of the Population Standard Deviation Use results from a prior survey Conduct a pilot survey Use secondary date Use judgment John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Problems Involving Proportions If there is no basis for estimating P, the researcher can make the most pessimistic case regarding the value of P Given the values of Z and E, what value of P will require the largest possible sample A value of .50 will make the value of the expression P(1-P) larger than any possible value of P. John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Population Size and Sample Size Make an adjustment in the sample size if the sample size is more than 5 percent of the size of the total population.  x =  n √ N - n N - 1 John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Determining Sample Size To recognize problems involving sampling means and proportions. Finite Population Correction (FPC) An adjustment in cases where the sample is expected to be equal to 5 percent or more of the total population. (N-n) / (N-1) nN N + n -1 n' = where: n' = revised sample size n = original sample size N = population size John Wiley & Son, Inc

To recognize problems involving sampling means and proportions. Statistical Power To recognize problems involving sampling means and proportions. Types of Errors Type I Error The error of concluding that there is a difference when there is no difference The formulas in this chapter focus on this type of error Type II Error The error of saying that there is no difference when there is a difference Statistical Power The probability of not making a Type II error John Wiley & Son, Inc

Determining Sample Size for Probability Samples SUMMARY Determining Sample Size for Probability Samples Methods for Determining Sample Size The Normal Distribution Population, Sample, and Sampling Distributions Sampling Distribution of the Mean Sampling Distribution of the Proportion Sample Size Determination Statistical Power John Wiley & Son, Inc

The End Copyright © 2006 John Wiley & Sons, Inc. John Wiley & Son, Inc