Chapter Thirteen Copyright © 2004 John Wiley & Sons, Inc. Sample Size Determination

Learning Objective John Wiley & Son, Inc 2 1.To learn the financial and statistical issues in the determination of sample size 2.To discover methods for determining sample size 3.To gain an appreciation of a normal distribution Learning Objectives

Learning Objective John Wiley & Son, Inc 3 4.To understand population, sample, and sampling distributions. 5.To distinguish between point and interval estimates 6.To recognize problems involving sampling means and proportions. Learning Objectives

Learning Objective John Wiley & Son, Inc 4 The financial and statistical issues in the determination of sample size. Determining Sample Size for Probability Samples 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.

Learning Objective John Wiley & Son, Inc 5 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

Learning Objective John Wiley & Son, Inc 6 Determining Sample Size for Probability Samples The financial and statistical issues in the determination of sample size. Number of Subgroups To Be Analyzed –The sample should contain at least 100 respondents in each major subgroup. 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.

Learning Objective John Wiley & Son, Inc 7 The Normal Distribution To gain an appreciation of a 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

Learning Objective John Wiley & Son, Inc 8 The Normal Distribution To gain an appreciation of a normal distribution. Important Characteristics of the Normal Distribution 1.The normal distribution is bell-shaped and has only one mode. 2.Symmetrical about the mean 3.Uniquely defined by its mean and standard deviation. 4.The total area is equal to one. 5.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. 6.The area between the mean and a given number of standard deviations from the mean is the same for all normal distributions

Learning Objective John Wiley & Son, Inc 9 The Normal Distribution To gain an appreciation of a 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.

Learning Objective John Wiley & Son, Inc 10 Z= value of the variable - mean of the variable standard deviation of the variable Z = X -   where X = value of the variable  = mean of the variable  = standard deviation of the variable The Normal Distribution To gain an appreciation of a normal distribution.

Learning Objective John Wiley & Son, Inc 11 To understand population, sample, and sampling distributions. Population and Sample Distributions Population Distribution –A frequency distribution of all the elements of a population. Sample Distribution –A frequency distribution of all the elements of an individual sample. Sampling Distribution of the Sample Mean –A frequency distribution of the means of many sample means from a given population

Learning Objective John Wiley & Son, Inc 12 s x =  n √ Sampling Distributions 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:

Learning Objective John Wiley & Son, Inc 13 Sampling Distributions 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

Learning Objective John Wiley & Son, Inc 14 The Standard Error of the Mean –Applies to the standard deviation of a distribution of sample means  x x =  n √ To understand population, sample, and sampling distributions. Sampling Distributions Of The Mean

Learning Objective John Wiley & Son, Inc 15 Sampling Distribution 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

Learning Objective John Wiley & Son, Inc 16 To distinguish between point and interval estimates. Sampling Distributions Of The Mean 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

Learning Objective John Wiley & Son, Inc 17 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 x1 x x<  <x+ 1 x1 x To distinguish between point and interval estimates. Sampling Distributions Of The Mean

Learning Objective John Wiley & Son, Inc 18 To recognize problems involving sampling means and proportions. Sampling Distribution of the Proportion 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

Learning Objective John Wiley & Son, Inc 19 To recognize problems involving sampling means and proportions. Sampling Distribution of the Proportion P (1-P) SpSp = n √ A relative frequency distribution of the sample proportions of a large number of random samples of a given size drawn from a particular population. 1.Approximates a normal distribution 2.The mean proportion is equal to the population proportion. 3.Standard error computed as:

Learning Objective John Wiley & Son, Inc 20 Problems Involving Means The formula for calculating the required sample size for problems that involve the estimation of a mean: n = Z 2  2 E2E2 where: Z = level of confidence expressed in standard errors  = population standard deviation E = acceptable amount of sampling error To recognize problems involving sampling means and proportions. Determining Sample Size

Learning Objective John Wiley & Son, Inc 21 Determining Sample Size To recognize problems involving sampling means and proportions. Four Methods Used to Deal with the Estimate of the Population Standard Deviation 1.Use results from a prior survey 2.Conduct a pilot survey 3.Use secondary date 4.Use judgment

Learning Objective John Wiley & Son, Inc 22 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.

Learning Objective John Wiley & Son, Inc 23 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 x =  n √ N - n √ N - 1

Learning Objective John Wiley & Son, Inc 24 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

Learning Objective John Wiley & Son, Inc 25 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 Statistical Power To recognize problems involving sampling means and proportions.

Learning Objective John Wiley & Son, Inc 26 Determining Sample Size for Probability Samples The Normal Distribution Population and Sample Distributions Sampling distribution of the Mean Determining Sample Size Statistical Power SUMMARY

Learning Objective John Wiley & Son, Inc 27 The End Copyright © 2004 John Wiley & Sons, Inc.

Learning Objective John Wiley & Son, Inc 28 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 SUMMARY

Learning Objective John Wiley & Son, Inc 29 The End Copyright © 2004 John Wiley & Sons, Inc.