Sample size Determination

Pritish Biswal

Utsho Chowdhary

Anurag Guha

Gopal Kumar

General Idea

Sample size is usually determined by the primary objective of trial.

Sample size calculation should be explicitly mentioned in the protocol.

Determining sample size

The estimate of the population standard deviation

The acceptable level of sampling errorThe desired confidence level

Methods

Unaided JudgmentAll You can AffordAverage Size for samples for similar studiesRequired Size per cellUse of a Traditional Statistical ModelUse of a Bayesian Model

Sampling Distribution

To help understand the concept of a sampling distribution of the mean by drawing samples from a population of 1250 sales invoices.

We use simple random sample of size n=50 from this population for the illustration.

A sampling distribution of the mean is the relative frequency distribution of the means of all possible samples of size n taken from a population of size N should be taken, and the mean of each sample be calculated and plotted in a relative frequency distribution.

sampling distribution of the meanFrequency of Sample Means

Relative frequency of sample means

38-39.99 1 1/500=.002

40-41.99 2 2/500=.004

42-43.99 17 17/500=.034

44-45.99 39 39/500=.078

46-47.99 52 52/500=.104

48-49.99 85 85/500=.170

50-51.99 110 110/500=.220

52-53.99 77 77/500=.154

54-55.99 64 64/500=.128

56-57.99 37 37/500=.074

58-59.99 10 10/500=.020

60-61.99 4 4/500=.008

62-63.99 2 2/500=.004

Total 500 1.000

Facts

A sampling distribution of the mean for simple random samples that are large(30 or more) hasA normal distributionA mean equal to the population(M)A standard deviation called the standard error

of the mean,i.e equal to the population standard deviation divided by the square root of the sample size

Statistical Estimation & the sampling distribution of the Mean

We want to estimate a population mean that we do not know from a sample mean. Two kinds of estimates of a population mean Point-An estimate involving only a single value. If a

random sample is taken, the sample mean is the best estimate that can be made from the sample data.

Interval-An estimate concerning an interval, or range of values. A statement of probability that the interval will enclose the true value of the mean is also given. It is called confidence coefficient and the interval is called a confidence interval.

Sampling distribution of the proportion

A sampling distribution of the proportion is the relative frequency distribution of the proportion(p) of all possible samples of size n taken from a population of size N.A sampling distribution of a proportion for a simple random sample has A normal distributionA mean equal to the population proportion(p)A standard error

Traditional statistical methods of determining sample size

What information is needed before a calculation of the sample size can be made?

Specification of error that can be allowed-how close must the estimate be?

Specification of confidence coefficient-what level of confidence is required that the actual sampling error does not exceed that specified?

Estimate of the population standard deviation-what is the s.d of the population?

Estimating variances for rating scales used in marketing research

On a 5 point scale responses can't be <1 or >5.This constraint leads to a relationship between mean &

variance. If a sample mean is 4.6 on a 5 point scale,then there

must be a large proportion of responses of 5. If the mean is near 3,the variance can be potentially

much greater.The nature of the relationship between the mean and

the variance depends on the number of scale points and on the shape of the distribution of responses.

Specification required for estimation problems involving proportions

The specification must be made to determine the sample size for an estimation problem involving a proportion are very similar to those for the mean

Specification of error that can be allowed-how close must the estimate be?

Specification of confidence coefficient-what level of confidence is required that the actual sampling error does not exceed that specified?

Estimate of population proportion using prior information-what is the approximate or estimated population proportion?

Sample size, Incidence & Nonresponse

Incidence-is the percentage of individuals who have the traits necessary to be included in a survey.

Nonresponse-refers to the percentage of respondents who refuse to participate in a survey or can’t be contacted.

Initial sample size=required response÷(incidence× response rate)

Thank You