Sample Size Determination Determining Sample Size for Probability

Sample Size Determination

Determining Sample Size for Probability Samples • Determining sample size for probability samples involves financial, statistical, and managerial considerations. • The larger the sample, the smaller the sampling error. In turn, the cost of the research grows with the size of sample.

Relationship Between Sample Size and Error 3

Determining Sample Size for Probability Samples • There are several methods for determining sample size. 1. To base the decision on the funds available. In essence, sample size is determined by the budget. Although seemingly unscientific, this approach is often a realistic one in the world of marketing research.

Determining Sample Size for Probability Samples • There are several methods for determining sample size. 2. The so-called rule of thumb method, which involves determining the sample size based on a gut feeling or common practice. The client often requests samples of 300, 400, or 500 in a request for proposal (RFP).

Determining Sample Size for Probability Samples • There are several methods for determining sample size. 3. A third technique is based on the number of subgroups to be analyzed. Generally speaking, the more subgroups that need to be analyzed, the larger the required total sample size.

Determining Sample Size for Probability Samples • In addition to these methods, there a number of statistical formulas for determining sample size. • Three pieces of data are required to make sample size calculation: – an estimate of the population standard deviation, – the level of sampling error that researcher/client is willing to accept, – and the desired level of confidence that the sample result will fall within a certain range of the true population value.

Normal Distribution • The normal distribution is crucial to statistical sampling theory. It is the bell-shaped and is symmetric about its mean; the mean, median, and mode are equal.

Standard Normal Distribution • The standard normal distribution has the features of a normal distribution; however, the mean of the standard normal distribution is always equal to zero, and the standard deviation is always equal to one.

Determining Sample Size • Problems involving means For example: the task estimating how many times the average fast-food restaurant user visits a fast-food restaurant in an average month.

Sample Size Formula: Problems involving means where: n = sample size z = confidence interval in standard error units s = standard error of the mean E = acceptable magnitude of error

Example #1 Suppose a survey researcher, studying expenditures on lipstick, wishes to have a 95% confident level (Z) and a range of error (E) of less than $2. 00. The estimate of the standard deviation is $29. 00. (z is the z-score, e. g. 1. 645 for a 90% confidence interval, 1. 96 for a 95% confidence interval, 2. 58 for a 99% confidence interval)

Calculation: Example #1

Example #2 Suppose, in the same example as the one before, the range of error (E) is acceptable at $4. 00. By how much is sample size is reduced?

Calculation: Example #2

Calculating Sample Size 99% Confidence (2. 57)( 29) ù é n=ê ú 2 ë û 2 é 74. 53 ù =ê ú ë 2 û 2 = [37. 265] = 1389 2 (2. 57)( 29) ù é n=ê ú 4 ë û 2 74. 53 ù é =ê ú 4 ë û 2 = [18. 6325] = 347 2

Determining Sample Size • Problems involving proportions For example: the problem of estimating the proportion/percentage of all adults who have accessed Twitter in the past 90 days. The goal is to take a simple random sample from the population of all adults to estimate this proportion.

Sample Size Formula: Problems involving proportions

Where: z 2 pq n= 2 E n = number of items in samples Z 2 = square of confidence interval in standard error units p = estimated proportion of success q = (1 -p) or estimated the proportion of failures E 2 = square of maximum allowance for error between true proportion and sample proportion, or zsp squared.

Calculating Sample Size at the 95% Confidence Level p =. 6 q =. 4 (1. 96 )2(. 6 )(. 4 ) n= (. 035 )2 ( 3. 8416 )(. 24 ) 001225. 922 =. 001225 = 753 =