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The rapid progression of emerging technologies like the Internet of Things (IoT) and Big Data analytics for manufacturing has driven innovation across various industries worldwide. Production data are utilized to construct a model for quality evaluation and analysis applicable to components processed by machine tools, ensuring process quality for critical components and final product quality for the machine tools. Machine tool parts often encompass several quality characteristics concurrently, categorized into three types: smaller-the-better, larger-the-better, and nominal-the-better. In this paper, an evaluation index for the nominal-the-better quality characteristic was segmented into two single-sided Six Sigma quality indexes. Furthermore, the process quality of the entire component product was assessed by n single-sided Six Sigma quality indexes. According to numerous studies, machine tool manufacturers conventionally base their decisions on small sample sizes (n), considering timeliness and costs. However, this often leads to inconsistent evaluation results due to significant sampling errors. Therefore, this paper established fuzzy testing rules using the confidence intervals of the q single-sided Six Sigma quality indices, serving as the fuzzy quality evaluation model for components of machine tools.

In this paper, estimation and prediction problems are discussed for the half-normal distribution under a progressively Type-II censoring scheme. This study focuses on two statistical inferential problems. In the first part of the study, several point estimators and confidence intervals are obtained for the scale parameter of the half-normal distribution. In the second part, several predictors and predictive intervals are derived for the removed failure times. A Monte Carlo simulation study is performed to discuss the mean squared error (mean squared prediction errors) and bias of estimates (predictors). The coverage probabilities and average length of the confidence and predictive intervals are simulated and a numerical example is provided.

When a researcher wants to perform a life-test comparison study of items made by two separate lines inside the same institution, joint censoring strategies are particularly important. In this paper, we present a new joint Type-Ⅰ hybrid censoring that enables an experimenter to stop the investigation as soon as a pre-specified number of failures or time is first achieved. In the context of newly censored data, the estimates of the unknown mean lifetimes of two different Rayleigh populations are acquired using maximum likelihood and Bayesian inferential techniques. The normality characteristic of classical estimators is used to offer asymptotic confidence interval bounds for each unknown parameter. Against gamma conjugate priors, the Bayes estimators and related credible intervals are gathered about symmetric and asymmetric loss functions. Since classical and Bayes estimators are acquired in closed form, simulation tests can be easily made to evaluate the effectiveness of the proposed methodologies. The efficiency of the suggested approaches is examined in terms of four metrics, namely: Root mean squared error, average relative absolute bias, average confidence length, and coverage probability. To demonstrate the applicability of the offered approaches to real events, two real applications employing data sets from the engineering area are analyzed. As a result, when the experimenter's primary goal is to complete the test as soon as the total number of failures or the threshold period is recorded, the numerical results reveal that the recommended strategy is adaptable and very helpful in completing the study.

Applying a statistical test to check statistical assumptions offers a positive or negative response regarding the veracity of the issued hypothesis. In case of variance analysis it’s necessary to apply a post hoc test to determine differences within the group. Statistical estimation using confidence levels provides more information than a statistical test, it shows the high degree of uncertainty resulting from small samples and builds conclusions in terms of "marginally significant" or "almost significant (p being close to 0,05) . The case study shows how the statistical estimation completes the application form the analysis of variance test and Tukey test.

As part of a research program, an extensive study on the dispersion characteristics of eight different 7.62 × 51 mm ammunition types was conducted. The paper presents the main steps in the experimental and analytical process carried out to evaluate, namely to measure and compare, the dispersion characteristics of the ammunitions; namely, (1) identify the number of rounds to fire in the trials, (2) establish a test plan and the setup for the precision trials, (3) fire the rounds, following an established protocol for the experiments, (4) collect the impact points, and measure the performance through statistical measures, (5) perform a statistical analysis of dispersion applied to the results obtained in the trials, and (6) conclude on the ammunition characteristics. In particular, the paper proposes a statistical method to evaluate the precision of ammunitions fired with precision (Mann) barrels. The practical method relies on comparison of confidence intervals and hypothesis testing on the standard deviation of samples, namely the impact points. An algorithm is proposed to compare the variances of two or more populations of ammunitions.

Power system operators, when obtaining a model's parameter estimates; require additional information to guide their decision on a model's acceptance. This information has to establish a relationship between the estimates and the chosen model in the parameter space. For this purpose, this paper proposes to extend the usage of the particle filter (PF) as a method for the identification of power plant parameters; and the parameters' confidence intervals, using measurements. Taking into consideration that the PF is based on the Bayesian filtering concept, the results returned by the filter contain more information about the model and its parameters than usually considered by power system operators. In this paper the samples from the multi-modal posterior distribution of the estimate are used to identify the distribution shape and associated confidence intervals of estimated parameters. Three methods [rule of thumb, least-squares cross validation, plug-in method (HSJM)] for standard deviation (bandwidth) selection of the Gaussian mixture distribution are compared with the uni-modal Gaussian distribution of the parameter estimate. The applicability of the proposed method is demonstrated using field measurements and synthetic data from simulations of a Greek power plant model. The distributions are observed for different system operation conditions that consider different types of noise. The method's applicability for model validation is also discussed.

A wide variety of approaches for set-valued simulation, parameter identification, state estimation as well as reachability, observability and stability analysis for nonlinear discrete-time systems involve the propagation of ellipsoids via nonlinear functions. It is well known that the corresponding image sets usually possess a complex shape and may even be nonconvex despite the convexity of the input data. For that reason, domain splitting procedures are often employed which help to reduce the phenomenon of overestimation that can be traced back to the well-known dependency and wrapping effects of interval analysis. In this paper, we propose a simple, yet efficient scheme for simultaneously determining outer and inner ellipsoidal range enclosures of the solution for the evaluation of multi-dimensional functions if the input domains are themselves described by ellipsoids. The Hausdorff distance between the computed enclosure and the exact solution set reduces at least linearly when decreasing the size of the input domains. In addition to algebraic function evaluations, the proposed technique is—for the first time, to our knowledge—employed for quantifying worst-case errors when extended Kalman filter-like, linearization-based techniques are used for forecasting confidence ellipsoids in a stochastic setting.

This paper is aimed at presenting application of bootstrap interval estimation methods to the assessment of financial investment’s effectiveness and risk. At first, we give an overview of various methods of bootstrap confidence interval estimation, i.e. bootstrap-t interval, percentile interval and BCa interval. Then, bootstrap confidence interval estimation methods are used to estimate confidence intervals for the Sharpe ratio and TailVaR of the Warsaw Stock Exchange sectoral indices. The results show that the bootstrap confidence intervals of different types are quite similarly positioned for each of the analysed index and measure. Taking into the account the locations of confidence intervals for both the Sharpe ratio and TailVaR, the real estate sector tends to be the most advantageous from the investor’s viewpoint.

This paper presents an implementation in R of the Cluster Estimated Standard Errors (CESE) proposed by [12]. The method estimates the covariance matrix of the estimated coefficients of linear models in grouped data sets with correlation among observations within groups. Cluster Estimated Standard Errors (CESE) is an alternative solution for the classical Cluster Robust Standard Errors (CRSE) [8, 13, 15], which underestimates the standard errors in most of the situations encountered in practice [7].

This piece is meant to help you understand and master two-level linear modeling in an accessible, swift, and fun way (while being based on rigorous and up-to-date research). It is divided into four parts: PART 1 presents the three key principles of two-level linear modeling. PART 2 presents a three-step procedure for conducting two-level linear modeling using SPSS, Stata, R, or Mplus (from centering variables to interpreting the cross-level interactions). PART 3 presents the results from a series of simulations comparing the performances of SPSS, Stata, R, and Mplus. PART 4 gives a Q&A addressing multilevel modeling issues pertaining to statistical power, effect sizes, complex design, and nonlinear two-level regression. The empirical example used in this tutorial is based on genuine data pertaining to ʼ90s and post-ʼ00s boy band member hotness and Instagram popularity. In reading this paper, you will have the opportunity to win a signed picture of Justin Timberlake.

Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.

Confidence intervals are useful tools for statistical decision-making purposes. In case of incomplete and vague data, fuzzy confidence intervals can be used for decision making under uncertainty. In this paper, we develop interval-valued intuitionistic fuzzy (IVIF) confidence intervals for population mean, population proportion, differences in means of two populations, and differences in proportions of two populations. The developed IVIF intervals can be used in cases of both finite and infinite population sizes. The developed fuzzy confidence intervals are equivalent decision-making tools to fuzzy hypothesis tests. We apply the proposed confidence intervals to the differences in the mean lives and failure proportions of two types of radiators used in automobiles, and a sensitivity analysis is given to check the robustness of the decisions.

Effect sizes and confidence intervals are important statistics to assess the magnitude and the precision of an effect. The various standardized effect sizes can be grouped in three categories depending on the experimental design: measures of the difference between two means (the $d$ family), measures of strength of association (e.$g$., $r$, $R^2$, $\eta ^2$, $\omega ^2$), and risk estimates (e.g., odds ratio, relative risk, phi; Kirk, 1996). Part I of this study reviews the $d$ family, with a special focus on Cohen's $d$ and Hedges' $g$ for two-independent groups and two-repeated measures (or paired samples) designs. The present paper answers questions concerning the $d$ family via Monte Carlo simulations. First, four different denominators are often proposed to standardize the mean difference in a repeated measures design. Which one should be used? Second, the literature proposes several approximations to estimate the standard error. Which one most closely estimates the true standard deviation of the distribution? Lastly, central and noncentral methods have been proposed to construct a confidence interval around $d$. Which method leads to more precise coverage, and how to calculate it? Results suggest that the best way to standardize the effect in both designs is by using the pooled standard deviation in conjunction with a correction factor to unbias $d$. Likewise, the best standard error approximation is given by substituting the gamma function from the true formula by its approximation. Lastly, results from the confidence interval simulations show that, under the normality assumption, the noncentral method is always superior, especially with small sample sizes. However, the central method is equivalent to the noncentral method when $n$ is greater than 20 in each group for a between-group design and when $n$ is greater than 24 pairs of observations for a repeated measures design. A practical guide to apply the findings of this study can be found after the general discussion.

This paper describes the classical and Bayesian estimation for the parameters of the Burr Type XII distribution based on generalized progressive Type I hybrid censored sample. We first discuss the maximum likelihood estimators of unknown parameters using the expectation-maximization (EM) algorithm and associated interval estimates using Fisher information matrix. We then derive the Bayes estimators with respect to different symmetric and asymmetric loss functions. In this regard, we use Lindley's approximation and importance sampling methods. Highest posterior density (HPD) intervals of unknown parameters are constructed as well. The results of simulation studies and real data analysis are conducted to compare the performance of the proposed point and interval estimators.

Most psychological research employs tables to report descriptive and inferential statistics. Unfortunately, those tables often misrepresent critical information on the shape and variability of the datas distribution. In addition, certain information such as the modality and score probability density is hard to report succinctly in tables and, indeed, not reported typically in published research. This paper discusses the importance of using graphical techniques not only to explore data but also to report it effectively. In so doing, the role of exploratory data analysis in detecting Type I and Type II errors is considered. A small data set resembling a Type II error is simulated to demonstrate this procedure, using a conventional parametric test. A potential analysis routine to explore data is also presented. The paper proposes that essential summary statistics and information about the shape and variability of data should be reported via graphical techniques.

Abstract Uncertainty estimates are important when retrieving properties of clouds and aerosols from satellites measurements. These measurements must be interpreted using a form of inverse theory, such as optimal estimation. In atmospheric remote sensing these inverse methods often assume that the forward model is linear in the region of uncertainty. This assumption is not necessarily valid. This paper presents an exact confidence procedure in contrast to the linear approximation using a maximum likelihood estimator. Two simple examples of retrieving the effective radius and optical depth of a volcanic ash cloud and water cloud show a discrepancy between the linear approximation and the exact procedure. The exact procedure is especially useful for inference where the entire parameter space has been forward modelled prior to or during the retrieval, such as using look up tables. When the inference method calculates the likelihood over the whole parameter space, it is less computationally expensive than a linear approximation.

This paper presents a review of Environmental Niche Modeling and a methodology for isolating the environmental niche of an aquatic species, given that prior information is available for characterising the physical tolerances for that species. To test and demonstrate our methodology, the environmental niche of the kelp bass has been isolated within Big Fisherman's Cove, Santa Catalina Island, CA, at specific confidence intervals. The motivation for this examination is to demonstrate the utility of ecological analysis in Robotics. Specifically, the utilisation of physical water properties to provide relative navigation and localisation for an aquatic robot. The environmental niches act as navigational landmarks in the seemingly featureless underwater environment. As water patches tend to stick together, this provides persistent landmarks for use in aquatic navigation problems. We provide the background and development of Environmental Niche Models, and present results from field trials for solving the navigation and localisation problem for underwater vehicles. Specifically, we provide results from a technique developed by the authors that utilises physical water parameters, e.g., temperature, salinity, chlorophyll, etc., to localise an underwater vehicle in a given region. The presented method leverages the concept of Environmental Niche Models to provide accurate position estimation that rivals GPS accuracy.

A common problem in displaying within-subject data is that of how to show confidence intervals that accurately reflect the pattern of significant differences between conditions. The Cousineau-Morey method \parencite {c05,m08} is a widely used solution to this issue; however, this method only applies to experimental designs with only one repeated-measures factor (e.g., subjects). Many experimental designs in fields such as psycholinguistics and social psychology use crossed random effect designs where, e.g., there are repeated measures both for subjects and stimuli. For such designs, extant methods for showing within-subject intervals would require first aggregating over stimuli, and thus such intervals might be a less accurate reflection of the statistical significance patterns if the data are actually analyzed using a method that takes both random effects into account (e.g., linear mixed-effects models). The present paper proposes an extension of the method described above to address this problem; the proposal is to scale the data using a mixed-effects model, rather than using the means from each subject, and then calculate confidence intervals from the data scaled thusly. Analysis of a sample of crossed random effect datasets reveals that intervals calculated using this method give a slightly more accurate reflection of the pattern of statistical significance in the between-condition differences.

The results of the method for estimating confidence intervals according to Steiger and Fouladi (1997) were not displayed correctly in our published paper (Figure 7 in Appendix C). We correct the figure here. The new figure changes our interpretation of the method to a more favorable appreciation.

The process of data mining with differential privacy produces results that are affected by two types of noise: sampling noise due to data collection and privacy noise that is designed to prevent the reconstruction of sensitive information. In this paper, we consider the problem of designing confidence intervals for the parameters of a variety of differentially private machine learning models. The algorithms can provide confidence intervals that satisfy differential privacy (as well as the more recently proposed concentrated differential privacy) and can be used with existing differentially private mechanisms that train models using objective perturbation and output perturbation.

Quantitative psychological research is focused on detecting the occurrence of certain population phenomena by analyzing data from a sample, and statistics is a particularly helpful mathematical tool that is used by researchers to evaluate hypotheses and make decisions to accept or reject such hypotheses. In this paper, the various statistical tools in psychological research are reviewed. The limitations of null hypothesis significance testing (NHST) and the advantages of using effect size and its respective confidence intervals are explained, as the latter two measurements can provide important information about the results of a study. These measurements also can facilitate data interpretation and easily detect trivial effects, enabling researchers to make decisions in a more clinically relevant fashion. Moreover, it is recommended to establish an appropriate sample size by calculating the optimum statistical power at the moment that the research is designed. Psychological journal editors are encouraged to follow APA recommendations strictly and ask authors of original research studies to report the effect size, its confidence intervals, statistical power and, when required, any measure of clinical significance. Additionally, we must account for the teaching of statistics at the graduate level. At that level, students do not receive sufficient information concerning the importance of using different types of effect sizes and their confidence intervals according to the different types of research designs; instead, most of the information is focused on the various tools of NHST.

Objectives: With the increasing emphasis on evidence-based medicine, the urology literature has seen a rapid growth in the number of high-quality randomized controlled trials along with increased statistical rigor in the reporting of study results. P-values, CI, and number needed to treat (NNT) are becoming increasingly common in the literature. This paper seeks to familiarize the reader with statistical measures commonly used in the evidence-based literature. Materials and Methods: The meaning and appropriate interpretation of these statistical measures is reviewed through the use of a clinical scenario. Results: The reader will be better able to understand such statistical measures and apply them to the critical appraisal of the literature. Conclusions: P-values, CI, and NNT each provide a slightly different estimate of statistical truth. Together, they provide a more complete picture of the true effect observed in a study. An understanding of these measures is essential to the critical appraisal of study results in evidence-based medicine.

This paper presents a frequentist comparison of the performance of confidence and credibility intervals for the difference of two proportions from two independent samples. The comparison is carried out considering three frequentist criteria. It was found that the intervals with the best performance, in terms of coverage probability, are Bayesians; in terms of expected length and variance of the length, the Newcombe interval shows the best performance. As a final remark, it was found that traditional intervals such as the Wald and adjusted Wald have a poor performance.Este artículo presenta una comparación del comportamiento de intervalos de confianza frecuentistas y de credibilidad bayesianos para la diferencia de dos proporciones provenientes de muestras aleatorias independientes. La comparación se lleva cabo considerando tres criterios frecuentistas con los cuales se concluyó que el mejor comportamiento, en términos de la probabilidad de cobertura, lo tienen los intervalos bayesianos, y en términos de la longitud esperada y varianza de la longitud el mejor comportamiento está dado por el intervalo frecuentista de Newcombe. Como resultado de esta investigación se encontró que los intervalos frecuentistas más populares como Wald y Wald ajustado tienen un comportamiento deficiente.

This study attempts to extend the prevailing definition of informational entropy, where entropy relates to the amount of reduction of uncertainty or, indirectly, to the amount of information gained through measurements of a random variable. The approach adopted herein describes informational entropy not as an absolute measure of information, but as a measure of the variation of information. This makes it possible to obtain a single value for informational entropy, instead of several values that vary with the selection of the discretizing interval, when discrete probabilities of hydrological events are estimated through relative class frequencies and discretizing intervals. Furthermore, the present work introduces confidence limits for the informational entropy function, which facilitates a comparison between the uncertainties of various hydrological processes with different scales of magnitude and different probability structures. The work addresses hydrologists and environmental engineers more than it does mathematicians and statisticians. In particular, it is intended to help solve information-related problems in hydrological monitoring design and assessment. This paper first considers the selection of probability distributions of best fit to hydrological data, using generated synthetic time series. Next, it attempts to assess hydrometric monitoring duration in a netwrok, this time using observed runoff data series. In both applications, it focuses, basically, on the theoretical background for the extended definition of informational entropy. The methodology is shown to give valid results in each case.

En el presente trabajo, los autores revisan algunos conceptos básicos sobre variables y procesamiento estadístico para obtener el valor "p" como el Test t de Student y el método de Chi-Cuadrado, así como la forma de calcular Intervalos de Confianza con las herramientas estadísticas del programa informático Excel (Microsoft), con ejemplos y un ejercicio para el lector.Authors of this paper review some basic concepts about variables and statistical processing to obtain "p" value, such as Student t-Test and Chi-Square test, as well as a way of obtaining Confidence Intervals with statistical tools from Excel software (Microsoft), providing examples and an exercise to the reader.

Different methodologies for obtaining confidence and credibility intervals for a proportion are studied in this paper. Expected coverage, length and length variance of the interval are defined and used as a means for comparing the intervals produced by each methodology presented. These indicators were calculated using the statistical package R, used to characterize each interval; furthermore, some known properties, such as Wald interval’s proportion undercoverage and Exact interval’s overcoverage are verified in this study.En este artículo se evalúa y se compara el comportamiento de diferentes metodologías empleadas para la obtención de intervalos de confianza de credibilidad, analizando sus probabilidades de cobertura estimada, su longitud esperada y la varianza de su longitud. Definidos estos tres conceptos, la comparación entre los intervalos considerados se desarrolla mediante procesos computacionales utilizando el paquete estadístico R. En este proceso, además de la verificación de conclusiones conocidas, como el mal comportamiento del intervalo de Wald y la sobrecobertura del intervalo exacto, se determinan, entre otros aspectos, características de los intervalos relacionadas con la variabilidad de su longitud.

En este artículo se evalúa y se compara el comportamiento de diferentes metodologías empleadas para la obtención de intervalos de confianza de credibilidad, analizando sus probabilidades de cobertura estimada, su longitud esperada y la varianza de su longitud. Definidos estos tres conceptos, la comparación entre los intervalos considerados se desarrolla mediante procesos computacionales utilizando el paquete estadístico R. En este proceso, además de la verificación de conclusiones conocidas, como el mal comportamiento del intervalo de Wald y la sobrecobertura del intervalo exacto, se determinan, entre otros aspectos, características de los intervalos relacionadas con la variabilidad de su longitud.Different methodologies for obtaining confidence and credibility intervals for a proportion are studied in this paper. Expected coverage, length and length variance of the interval are defined and used as a means for comparing the intervals produced by each methodology presented. These indicators were calculated using the statistical package R, used to characterize each interval; furthermore, some known properties, such as Wald intervals proportion undercoverage and Exact intervals overcoverage are verified in this study.

El coeficiente alfa de Cronbach es el modo más habitual de estimar la fiabilidad de pruebas basadas en Teoría Clásica de los Test. En dicha estimación, los investigadores usualmente omiten informar intervalos de confianza para el coeficiente, un aspecto no solo recomendado por los especialistas, sino también requerido explícitamente en las normas editoriales de algunas revistas especializadas. Esta situación puede atribuirse a que los métodos de estimación de intervalos de confianza son poco conocidos, además de no estar disponibles al usuario en los programas estadísticos más populares. Así, en este trabajo se presenta un programa desarrollado dentro del sistema estadístico ViSta que permite calcular intervalos de confianza basados en el enfoque clásico y mediante la técnica bootstrap. Se espera promover la inclusión de intervalos de confianza para medidas de fiabilidad, facilitando el acceso a las herramientas necesarias para su aplicación. El programa es gratuito y puede obtenerse enviando un mail de solicitud al autor del trabajo.Cronbach's alpha coefficient is the most popular way of estimating reliability in measurement scales based on Classic Test Theory. When estimating it, researchers usually omit to report confidence intervals of this coefficient, as it is not only recommended by experts, but also required by some journal's guidelines. This situation is because the different methods of estimating confidence intervals are not well-known by researchers, as well as they are not being available among the most popular statistical packages. Therefore, this paper describes a computer program integrated into the ViSta statistical system, which allows computing confidence intervals based on the classical approach and using bootstrap technique. It is hoped that this work promotes the inclusion of confidence intervals of reliability measures, by increasing the availability of the required computer tools. The program is free and it can be obtained by sending an e-mail to the author.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations - something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

The need for accurate quantification of the amount of carbon stored in the environment has never been greater. Carbon sequestration has become a vital component of the battle against global climate change, and monitoring and quantifying this process are major challenges for policymakers. Plant allometric equations allow managers and scientists to quantify the biomass contained in a tree without cutting it down, and therefore can play a pivotal role in measuring carbon sequestration in forests and savannahs. These equations have been available since the beginning of the 20th century, but their usefulness depends on the ability to estimate the error associated with the equations – something which has received scant attention in the past. This paper provides a method based on the theory of linear regression and the lognormal distribution to derive confidence limits for estimates of biomass derived from plant allometric equations. Allometric equations for several southern African savannah species are provided, as well as the parameters and equations required to calculate the confidence intervals. This method was applied to data collected from a sampling campaign carried out in a savannah landscape at the Skukuza flux site, Kruger National Park, South Africa. Here the error was 10% of the total site biomass for the woody biomass and 2% for the leaf biomass. When the data were split into individual plots and used to estimate site biomass (as would occur in most sampling schemes) the error increased to 16% and 12% of the woody and leaf biomasses, respectively, as the sampling errors were added to the errors in the allometric equation. These methods can be used in any discipline that applies allometric equations, such as health sciences and animal physiology.

Boxplots are a useful and widely used graphical technique to explore data in order to better understand the information we are working with. Boxplots display the first, second and third quartile as well as the interquartile range and outliers of a data set. The information displayed by the boxplot, and most of its variations, is based on the data’s median. However, much of scientific applications analyse and report data using the mean. In this paper, we propose a variation of the classical boxplot that displays information around the mean. Some information about the median is displayed as well.

Confidence intervals and measures of effect size are gradually becoming the standard way of reporting the results of statistical analyses in research articles, used instead of or in addition to p values. However, this shift in research practices barely affected teaching practices up to now. This paper is the third of a series written to serve as a general reference on the use of confidence intervals in quantitative social sciences. Its purpose is to provide guidelines, advices and useful tricks of the trade that will allow readers (a) to face most of the statistical problems emerging in real-life research settings and (b) to improve their understanding of confidence intervals and answer more efficiently their questions of interest. The first part of the article briefly introduces the basic elements of an approach based on confidence intervals: Calculations, interpretation, and hypothesis testing. The second part is an attempt to present some of the most important (but sometimes neglected) advanced issues concerning confidence intervals: Graphic representations, complex distributions, national surveys, the larger family of interval statistics (e.g., prediction intervals), and the Bayesian approach to probabilities.

The constraints of conducting evaluations in real-world settings often necessitate the implementation of less than ideal designs. Unfortunately, the standard method for estimating the precision of a result (i.e., confidence intervals [CI]) cannot be used for evaluative conclusions that are derived from multiple indicators, measures, and data sources, for example. Moreover, CIs ignore the impact of sampling and measurement error. Considering that the vast majority of evaluative conclusions are based on numerous criteria of merit that often are poorly measured, a significant gap exists with respect to how one can estimate the CI of an evaluative conclusion. The purpose of this paper is (1) to heighten reader consciousness about the consequences of utilizing a weak evaluation design and (2) to introduce the need for the development a methodology that can be used to characterize the precision of an evaluative conclusion.

Abstract Introduction: The aim of this article is to study the efficacy and safety of cardiac shock wave therapy (CSWT) in the treatment of coronary heart disease (CAD). Methods: A comprehensive search of electronic databases and a manual search of conference papers and abstracts were performed until September 30, 2018. The studies using RevMan 5.3 and STATA 14.0 softwares were reviewed, and meta-analyses were performed on 13 indicators, such as a six-min walking distance test (6MWT), New York Heart Association (NYHA) functional class, Seattle Angina Questionnaire (SAQ) score, angina class (Canadian Cardiology Society [CCS]), etc. Results: A total of 26 articles were included. The total patient population was 855, of which 781 patients were treated with CSWT. Meta-analyses indicated that 6MWT (mean difference [MD] 75.64, 95% confidence interval [CI] 49.03, 102.25, P