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Experimentation
Published in Andrew Greasley, Simulation Modelling, 2023
A hypothesis test makes an assumption or a hypothesis (termed the null hypothesis, H0) and tries to disprove it. Acceptance of the null hypothesis implies that there is insufficient evidence to reject it (it does not prove that it is true). Rejection of the null hypothesis, however, means that the alternative hypothesis (H1) is accepted. The null hypothesis is tested using a test statistic (based on an appropriate sampling distribution) at a particular significance level α, which relates to the area called the critical region in the tail of the distribution being used. If the test statistic (which we calculate) lies in the critical region, the result is unlikely to have occurred by chance, and so the null hypothesis would be rejected. The boundaries of the critical region, called the critical values, depend on whether the test is two-tailed (we have no reason to believe that a rejection of the null hypothesis implies that the test statistic is either greater or less than some assumed value) or one-tailed (we have reason to believe that a rejection of the null hypothesis implies that the test statistic is either greater or less than some assumed value).
Sampling and estimation theories
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
The standard deviation of a sampling distribution of mean values is called the standard error of the means, thus standard error of the means, σx¯=σNNp−NNp−1
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.
Evaluating students’ conceptual and procedural understanding of sampling distributions
Published in International Journal of Mathematical Education in Science and Technology, 2019
Zeynep Medine Ozmen, Bulent Guven
Sampling distribution is defined as: The distribution of all values of the statistic when all possible samples of a sample size n are taken from the same population … The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table, probability histogram, or formula. [8,p.276]
Concurrent optimization of parameter and tolerance design based on the two-stage Bayesian sampling method
Published in Quality Technology & Quantitative Management, 2023
Yan Ma, Jianjun Wang, Yiliu Tu
The loss function has been widely used in quality improvement activities. It can effectively measure the loss to society due to product quality problems after the product is delivered to the customers (Wang et al., 2016). In addition to the univariate loss function originally proposed by Taguchi, different multivariate loss functions have been proposed by several researchers (Ko et al., 2005; Pignatiello, 1993; Vining, 1998; Wang et al., 2016). Most of these loss functions assume that their responses follow a normal distribution. However, this assumption does not hold in many industrial experiments. This article focuses on the IPTD considering non-normal multiple responses. Therefore, the proposed loss function in this article should not only apply to normal responses but also non-normal responses. We propose a sampling-based loss function by combining the Bayesian sampling technique and central limit theorem (CLT) to solve the above problems. The idea of the loss function proposed in this paper is described as follows. First, a large number of posterior sampling values for each response can be obtained by the two-stage Bayesian sampling method described in Section 4.1. The CLT states that the sampling distribution of the response mean follows a normal distribution as the size of the sample increases, regardless of its original population distribution. According to the CLT, these response sampling values which fall within the specification limits will follow a multivariate normal distribution. Also, the mean vector and the variance-covariance matrix can be obtained by using these posterior samples . Referring to Vining (1998), the expected loss function for multiple responses can be given by