Downloaded from https://www.everyspec.com ADS-79D-HDBK deviation or population proportion. The required precision for the estimate should also be defined. In general, the potential use of the estimate, in terms of criticality of application, affects the precision requirement which further impacts sample size selection. A probability statement connecting the desired precision of the estimate with the sample size is the essential step in sample size determination. The statement may contain unknown properties of the population such as the mean or variance. This is where prior information can help. The final sample size should be scrutinized for practicality.
If it is unacceptable, the only way to reduce it is to accept less precision in the sample estimate. I.3.2 Concept of Confidence Interval and Hypothesis Testing. In statistics, a confidence interval is a particular kind of interval estimate of a population parameter. Instead of estimating the parameter by a single value, an interval is likely to include the given parameter. Thus, confidence intervals are used to indicate the reliability of an estimate. The width of these confidence intervals is a measure of the overall quality of the estimated parameter or regressed model.
A narrower confidence interval indicates a tighter statistical estimate of the parameter with less variability or standard deviation around its estimated value. Mathematically, a confidence interval can be defined as the boundary in which an experimental outcome is anticipated to stay for a given level of probability. In many engineering applications, it is often required that we make a decision whether to accept or reject a statistical statement about a statistical parameter of interest. The statement is often referred to as the hypothesis and the associated decision making procedure about the hypothesis is called hypothesis testing.
This is one of the most useful aspects of statistical inference, since many types of decision- making problems, tests, or experiments in the engineering world can be formulated as hypothesis-testing problems. Furthermore, there is a very close connection between hypothesis testing and confidence intervals. A statistical hypothesis is a statement about the parameters of one or more populations. For any given statement about parameter μ , there are two hypotheses: H 0 and H 1 , such as: where H 0 is called the Null Hypothesis and H 1 is referred to as the Alternative Hypothesis. From a probabilistic standard point of view, the chance that μ equals exactly μ 0 is zero.
Therefore, the Null Hypothesis is practically associated with a region, within which μ is close enough to μ 0 so the null hypothesis will be accepted. By convention, this region is usually called the acceptance region. In general, the confidence interval is used to determine the boundary for the acceptance region. Selection of a Null Hypothesis varies from one application to another. For the case associated with structural damage detection, the Null Hypothesis usually states that damage exists. For mechanical diagnostics, the Null Hypothesis typically states that no defect or unacceptable fault exists. I.3.3 Type I and Type II Error .
Due to the inherent variability associated with any experimental study, the outcome of an observable event exhibits some randomness. As a result, the aforementioned decision procedure may lead to two wrong conclusions: Type I error and Type II error. As listed in Table I-I, Type I error is related to cases where Non Destructive Equipment (NDE) or a CBM monitoring CI/HI indicates detection when no fault actually exists (false positive on a healthy component). While, Type II error is associated with the situations in which NDE or a CBM monitoring CI/HI misses detection when a fault exists (false negative or missed alarm on an unhealthy component). 148