### Power and Sample Size

Power and Sample Size analysis is useful for researchers to design their experiments. It can compute the power of the experiment for a given sample size, and can also compute the required sample size for given power values.

Power and sample size computations are test-dependent. That is, if the initial hypothesis test is a one-sample t-test, then the power and sample size computations must be based on that test, otherwise, the results may be incorrect.

The following four examples show how to perform power and sample size analysis for one-sample t-test, two-sample t-test, paired sample t-test and ANOVA. Each presents a realistic problem and shows how one would use the corresponding dialog in OriginPro to solve it.

1. Power and Sample Size for One-Sample t-Test
 Sociologists would like to determine whether or not the average infant mortality rate in the United States is equal to 8%. It is determined that the design difference of the rate cannot vary more than 0.5%. The researcher knows the standard deviation should be 2.1 in pilot studies. How many samples must be taken to estimate the average infant mortality rate at a confidence level of 95% (Alpha=0.05) for power values of 0.7, 0.8 and 0.9? In this case, power and sample size for one-sample t-test should be used.

1. Power and Sample Size for Two-Sample t-Test
 A doctor's office participates in two local insurance plans, Healthwise and Medicare. They want to compare the mean time (in days) until the reimbursement of claims for the two plans. Historical data shows that for the Healthwise plan, the average time is 32 days and standard deviation is 7.05 days. For the Medicare plan, the average reimbursement time is 42 days and the standard deviation is 3.5 days. If 5 claims from each plan were selected and corresponding reimbursement times were recorded, what is the power to detect the difference in mean reimbursement times between the 2 plans by 5% or more? In this case, power and sample size for two-sample t-test should be used. Please note that the value of Standard Deviation is calculated by

1. Power and Sample Size for Paired-Sample T-Test
 A production manager would like to know whether two ice cream packaging machines produce packages of equivalent volume. Two ice cream packaging machines are run parallel to each other with ice cream fed from one large vat. The production manager would like to do an experiment where he collects packages at different times across 2 days of production to determine if there is any difference in the average volume in the two packaging machines. The volume difference between the two machines cannot be more than 0.5 oz for 64 oz containers. So if a difference between the machines is 0.5 oz or larger we want to be able to detect it with a certain amount of certainty. (This will be the power.) It was decided to do this experiment on our special chocolate-marshmallow ice cream because at times there has been concern with the overall variability in the package volume and some have hypothesized that a major source of variation could be coming from these two packaging machines. It is well known that there is variation in the ice cream density over time. This could be causing the variation that people are noticing and not the machines themselves. So we expect to see some variation in packaging volume over time. Thus a paired approach will be needed. A similar experiment was done last year on vanilla ice cream. At that time no significant different between the two packaging machines was detected and a standard deviation of the difference was found to be 1 oz. So we will use this estimate of the standard deviation of the differences to plan this experiment. How many samples must be taken at a confidence level of 99% for power values of 0.8, 0.9, 0.95? In this case, power and sample size for paired-sample t-test should be used.

1. Power and Sample Size for One-Way ANOVA
 Researchers are interested in whether different plants have different nitrogen content. They plan to record nitrogen content in milligrams of 4 kinds of plants (20 observations per kind of plant). Previous research suggests the square root of MSE is 60 and the CSS of mean should be 400. Is the plan feasible? In this case, power and sample size for one-sample t-test should be used.