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Statistical Inference I
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
Despite the existence of the median, by far the most popular and useful measure of central tendency is the arithmetic mean, or, more succinctly, the sample mean or expectation. The sample mean is another statistical term that measures the central tendency, or average, of a sample of observations. The sample mean varies across samples and thus is a random variable. The mean of a sample of measurements x1, x2, …, xn is defined as () MEAN(X)=E[X]=X¯=∑i=1nxin,
Probability Distributions
Published in Alan R. Jones, Probability, Statistics and Other Frightening Stuff, 2018
For those curious to the reason, or just don’t like loose ends: As stated in the previous paragraph, the Central Limit Theorem espouses that the Sample Mean will be Normally Distributed about the Population Mean. Suppose that the process under consideration (e.g. a machining process) is required to produce components that are accurate to a specified design tolerance. Suppose further that the process is producing components with a Mean value that is one and a half Standard Deviations ‘out’ from the required specification due to machine wear, say, but are still within the required tolerance. In other words, the Population Mean has shifted by 1.5σ. The process will still be producing components that are Normally Distributed with a 6-Sigma spread – but with the wrong Mean. In order to achieve a rejection rate of less than 1 in 3.4 million we need to consider a span of 9σ nine Standard Deviations (see Figure 4.8.)
Sensor specification and testing tools
Published in Lawrence A. Klein, ITS Sensors and Architectures for Traffic Management and Connected Vehicles, 2017
For example, suppose it is desired to find a level C confidence interval for the mean μ of a population from an unbiased random data sample of size n. The confidence interval is based on the sampling distribution for the sample mean x¯, which is equal to N(μ,σ/n) when the sample is obtained from a population having the N(μ, σ) distribution. In this notation, N represents a normal distribution, μ the mean of the entire population, and σ the standard deviation of the entire population. The central limit theorem confirms that a normal distribution is a valid representation of the sampling distribution of the sample mean when the sample size is sufficiently large regardless of the probability density function that describes the statistics of the entire population.
Contribution to improvement of knowledge management in the construction industry - Stakeholders’ perspective on implementation in the largest construction companies
Published in Cogent Engineering, 2022
António Joaquim Coelho Marinho, João Couto
The Central Limit Theorem (CLT) states that the sampling distribution of a sample mean is approximately normal if the sample size is large enough, even if the population distribution is not normal. Therefore, to evaluate the adherence to normality, i.e., to test whether the sample can be assumed to be normal, it was used the Shapiro-Wilk (S-W) test, for it allows to determine whether a given sample comes from a population with normal distribution. The rejection or acceptance of normality is found by determining the critical value of the distribution of S-W statistics. For this purpose, it was calculated the hypothesis p-value (the probability of obtaining a test statistic equal to or greater than that observed in a sample), and if the p-value is less than or equal to α (probability of error), the normality of the sample must be rejected. The level α chosen was 0.05 for all tests. On the other hand, to test the null hypothesis (“H0: Is there any difference in the answers stemming from the respondents’ function in their companies?”), it was verified if the normality is observed.
Regional phase angle, not whole-body, is augmented in response to pre-season in professional soccer players
Published in Research in Sports Medicine, 2022
Renêe de Caldas Honorato, Alex Soares Marreiros Ferraz, Witalo Kassiano, Priscila Custódio Martins, Diego Augusto Santos Silva, Vânia Marilande Ceccatto
BIVA analysis. From the values of BIVA, BIVA confidence and BIVA tolerance were created. The BIVA confidence comprises the 95% confidence ellipses for the vector means that are found by averaging the components of the relationship between R and Xc by height (metres) measured in a group of individuals. The sample mean is presented as an estimate of the results that would be obtained if the total population was studied. Confidence intervals are used to check whether a mean is significantly different from a hypothetical value or a comparison population (Piccoli & Pastori, 2002). BIVA tolerance is the graphical analysis of the individual or of the three ellipses: the median, the third quartile, and the 95th percentile, which are regions that include 50%, 75%, and 95% of the individual points, respectively. In this way, the tolerance graph allows a more detailed classification of the vector position of the individual impedance (a point) in the R/Xc graph, through its distance concerning the mean vector of the reference population (Piccoli & Pastori, 2002) and with a population of university soccer athletes (Martins et al., 2021). In vector BIVA, standardization of Z by the conductor length allows a consistent evaluation of Z-body and Z-leg with the same electric unit in ohms per metre.
An efficient random-sampling method for calculating double occupancy of Gutzwiller wave function in single-band 1D and 2D lattices
Published in Molecular Physics, 2021
Feng Zhang, Zhuo Ye, Yong-Xin Yao, Cai-Zhuang Wang, Kai-Ming Ho
We calculate d numerically based on a system with finite N. It is clear in Equation (8) that all computational burden arises from calculating . has the form of an ensemble partition function, which in general cannot be effectively sampled in a single simulation. However, for a finite N, the total number of terms () contained in is also finite. For example, for a non-magnetised system with total electrons. Thus, to obtain , one only needs to sample the mean value , which is done by randomly sampling M configurations containing m double-occupied sites according to a combinatorial distribution. The sample mean is taken as an estimator of the population mean. is expected to converge much faster than itself since the standard deviation of the mean value of a random sample with size M scales as . A separate sampling is performed for each value of m. Then, the double occupancy d can be calculated for any g according to Equation (8).