Null Hypothesis

Why We Dont Accept the Null Hypothesis by Keith M. Bower, M. S. and James A. Colton, M. S. Reprinted with permission from the American Society for Quality When performing statistical hypothesis assays such as a one-sample t-test or the AndersonDarling test for normality, an investigator will either lower or fail to reject the zippo hypothesis, base upon sampled data. Frequently, results in Six Sigma projects contain the verbiage accept the empty hypothesis, which implies that the worthless hypothesis has been proven true.This article discusses wherefore such a practice is incorrect, and why this issue is more than a matter of semantics. Overview of Hypothesis Testing In a statistical hypothesis test, two hypotheses ar evaluated the null (H0) and the resource (H1). The null hypothesis is assumed true until proven otherwise. If the weight of evidence leads us to believe that the null hypothesis is highly un presumable (based upon prospect theory), then we sacrifice a statist ical basis upon which we may reject the null hypothesis. A common misconception is that statistical hypothesis tests are designed to select the more likely of two hypotheses.Rather, a test will stay with the null hypothesis until enough evidence (data) appears to support the alternative. The amount of evidence required to prove the alternative may be stated in terms of a confidence level (denoted X%). The confidence level is often specified before a test is conducted as part of a sample size calculation. We view the confidence level as equaling one minus the Type I erroneousness rate (? ). A Type I error is committed when the null hypothesis is wrongly rejected. An ? value of 0. 05 is typically employ, corresponding to 95% confidence levels.The p-value is used to determine if enough evidence exists to reject the null hypothesis in favor of the alternative. The p-value is the probability of incorrectly rejecting the null hypothesis. The two possible conclusions, after assessing the data, are to 1. Reject the null hypothesis (p-value ? ) and conclude that there is not enough evidence to state that the alternative is true at the pre-determined confidence level of X%. Note that it is possible to state the alternative to be true at the lower confidence level of 100*(1 p-value)%. Ronald A.Fisher succinctly discusses the key point of our reputation In relation to any experiment we may speak of the null hypothesis, and it should be noted that the null hypothesis is never proved or established, solely is possibly disproved, in the course of experimentation. Every experiment may be said to exist only in order to give the facts a take place of disproving the null hypothesis. 1 A Helpful Analogy The U. S. Legal System Consider the example of the legal system in the United States of America. A somebody is considered innocent until proven guilty in a court of law.We may state this particular decision-making process in the form of a hypothesis test, as follows H0 Person is innocent vs. H1 Person is not innocent (i. e. , guilty) The responsibility then falls upon the prosecution to build a case to prove guilt beyond a reasonable doubt. It should be borne in mind that a jury will never find a person to be innocent. The defendant would be found not guilty in such a situation i. e. , the jury has failed to reject the null hypothesis. Decisions Based on Data We must curb in mind, of course, that it is always possible to draw an incorrect conclusion based upon sampled data.There are two kinds of error we can make Type I error. When the null hypothesis is rejected, practitioners identify to the Type I error when they present results, using language such as We reject the null hypothesis at the 5% significance level, or We reject the null hypothesis at the 95% confidence level. Type II error. A second possible mistake involves incorrectly failing to reject the null hypothesis. The power of a test is defined as one minus the Type II error rate, and is therefore the probability of correctly rejecting H0. The sample size plays an important role in determining the statistical power of a test.When statisticians address small sample sizes, they often refer to the power to justify their concerns. One may argue that the sample size would be too low to correctly detect a inequality from the hypothesized value, if that difference truly existed. Example of a Test with Low Power Consider a test that compares the mean of a process to a target value. The null and alternative hypotheses are, respectively H0 Process mean on target vs. H1 Process mean different from target Suppose two observations are collected everyday to monitor for a change in the process mean (i. e. , n = 2). Assume a one-sample t-test is carried out at the ? 0. 05 significance level (95% confidence level) and the resulting p-value is supra 0. 05. Fig. 1 One-Sample t-Test As is shown in Figure 1, there is less than a 50% chance (power = 0. 4944) such a test will correctly reject the null hypothesis even when the difference between the process mean and the target is six standard deviations. This is obviously an enormous statistical difference, yet the test (owing to the small sample size) would not be sensitive to it. The danger in concluding the process is on target with a sample size of two, for this example, is evident. ImplicationsAssessing and relaying findings in a cogent manner is critical for Six Sigma practitioners. In statistical hypothesis testing procedures, this means that investigators should avoid misleading language such as that which implies acceptance of the null hypothesis. Reference th 1. Ronald A. Fisher, The Design of Experiments, 8 ed. (New York Hafner Publishing Company Inc. , 1966), 17. Bibliography 1. Lenth, Russell V. Some Practical Guidelines for Effective Sample Size Determination. The American mathematical statistician 55, no. 3 (2001) 187-193. 2. Tukey, John W. Conclusions vs. Decisions. Technometrics 2, no. 4 (1960) 423433.