The probability of a Type II error is denoted β, and β = P(Do not Reject H 0 | H 0 is false), i.e., the probability of not rejecting the null hypothesis if the null hypothesis were true. The critical value (93.92) is indicated by the vertical line. In our test, we selected α = 0.05 and reject H 0 if the observed sample mean exceeds 93.92 (focusing on the upper tail of the rejection region for now). If the true mean is 94, then the alternative hypothesis is true. The figure below shows the distributions of the sample mean under the null and alternative hypotheses.The values of the sample mean are shown along the horizontal axis.ĭistribution of Under H 0: μ = 90 and Under H 1: μ = 94 Now, suppose that the alternative hypothesis, H 1, is true (i.e., μ ≠ 90) and that the true mean is actually 94. This concept was discussed in the module on Hypothesis Testing. The areas in the two tails of the curve represent the probability of a Type I Error, α= 0.05. Rejection Region for Test H 0: μ = 90 versus H 1: μ ≠ 90 at α =0.05 The rejection region is shown in the tails of the figure below. Note that there is an alternative formula for estimating the mean of a continuous outcome in a single population, and it is used when the sample size is small (n 93.92. In practice we use the sample standard deviation to estimate the population standard deviation. In the module on confidence intervals we derived the formula for the confidence interval for μ as Confidence intervals for every parameter take the following general form: The module on confidence intervals provided methods for estimating confidence intervals for various parameters (e.g., μ, p, ( μ 1 - μ 2 ), μ d, (p 1-p 2)). Issues in Estimating Sample Size for Confidence Intervals Estimates Compute the sample size required to ensure high power when hypothesis testing.Interpret statistical power in tests of hypothesis.

Compute the sample size required to estimate population parameters with precision.Provide examples demonstrating how the margin of error, effect size and variability of the outcome affect sample size computations.Just as it is important to consider both statistical and clinical significance when interpreting results of a statistical analysis, it is also important to weigh both statistical and logistical issues in determining the sample size for a study.Īfter completing this module, the student will be able to: These financial constraints alone might substantially limit the number of women that can be enrolled. Suppose that the collection and processing of the blood sample costs $250 per participant and that the amniocentesis costs $900 per participant. The amniocentesis is included as the gold standard and the plan is to compare the results of the screening test to the results of the amniocentesis. In order to evaluate the properties of the screening test (e.g., the sensitivity and specificity), each pregnant woman will be asked to provide a blood sample and in addition to undergo an amniocentesis. Suppose that the screening test is based on analysis of a blood sample taken from women early in pregnancy. For example, suppose a study is proposed to evaluate a new screening test for Down Syndrome. However, in many studies, the sample size is determined by financial or logistical constraints. The formulas presented here generate estimates of the necessary sample size(s) required based on statistical criteria. Studies that are much larger than they need to be to answer the research questions are also wasteful. These situations can also be viewed as unethical as participants may have been put at risk as part of a study that was unable to answer an important question. Studies that have either an inadequate number of participants or an excessively large number of participants are both wasteful in terms of participant and investigator time, resources to conduct the assessments, analytic efforts and so on.

Studies should be designed to include a sufficient number of participants to adequately address the research question. This module will focus on formulas that can be used to estimate the sample size needed to produce a confidence interval estimate with a specified margin of error (precision) or to ensure that a test of hypothesis has a high probability of detecting a meaningful difference in the parameter. Boston Univeristy School of Public HealthĪ critically important aspect of any study is determining the appropriate sample size to answer the research question.

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