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Product Manufacturing Degradation Control
Published in Douglas Brauer, John Cesarone, Total Manufacturing Assurance, 2022
The Central Limit Theorem states that, if n is large enough, the distribution of the sample means will be Normally distributed. The only question left to be answered is, how large is large enough? The precise answer to this question depends on the precise distribution of the original data, and how close we wish to come to a Normal Distribution. The larger the n, the closer we come to true normality. In most engineering applications, four is considered a lower limit on sample size. A sample size of five is common in practice. However, if data are plentiful and measurements are cheap to make, ten is an even better size to use. The sample size in any actual implementation will be a decision based on the cost and availability of the data, and the precision required in the analysis.
Distributions
Published in Robert M. Bethea, R. Russell Rhinehart, Applied Engineering Statistics, 2019
Robert M. Bethea, R. Russell Rhinehart
Regardless of the shape of the distribution of the original population, the central limit theorem allows us to use the normal distribution for descriptive purposes, subject to a single restriction. The theorem simply states that if the population has a mean μ and a finite variance σ2, then the distribution of the sample mean X¯ approaches the normal distribution with mean μ and variance σ2/n as the sample size n increases. The chief problem with the theorem is how to tell when the sample size is large enough to give reasonable compliance with the theorem. The selection of sample sizes is covered in Chapter 9.
Probability Distributions
Published in Alan R. Jones, Probability, Statistics and Other Frightening Stuff, 2018
The Central Limit Theorem states that the Arithmetic Mean of a sufficiently large number of independent samples drawn from independent random variables with the same Population Mean and Population Variance will be Normally Distributed around the Population Mean. (For smaller sample sizes, refer to the Student’s t-Distribution in Chapter 6.) It is generally accepted that for most situations a ‘large sample size’ is taken to be greater than 30 observations (so, not that large in the scheme of things until we start to look for relevant data on non-recurring activities or events, in which case it can seem to become unachievably large! Oops, there goes my cynicism showing again!) This is an important result for estimators as it allows us to assess the sensitivity of an estimate generated through sampling methods.
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.
A Bayesian method for estimating uncertainty in excavated material
Published in International Journal of Mining, Reclamation and Environment, 2022
The data used in this paper were collected from a single grade block, thus the material transported to dump locations are spatially correlated. The same proposed method that used to infer the truck material was used for estimating the mean and uncertainty of Fe wt% at dump locations (Figure 10b). In order to infer the moments at the dump locations, the model assumed that the trucks are continuously arrived at the dump locations. Hence the model aggregates the continuous truck loads to the destination and the same spatial correlation applied. As seen in Figure 10b, the high uncertainty at the dump location F is due to the high uncertain truck loads that were loaded near high risk region. However, the uncertainty aggregated from the grade blocks are continuously significantly decreasing at trucks and then at dump locations (Figure 11). This is because, that the uncertainty of Fe wt% for a given bucket is estimated from the random sample locations, say n, and then the uncertainty of Fe wt % for a given truck is estimated from multiple buckets, say m, thus the samples in the truck are coming from n × m simulated locations. According to central limit theorem, as the sample size grows, the average can take more values and therefore the standard deviation gets smaller. Hence, in Figure 11, the mean and the standard deviation in trucks and dump locations converge.
How does older and younger drivers’ risk cognition affect the safety performance: A driving simulator study of sudden lane-changing of the leading vehicle
Published in Journal of Transportation Safety & Security, 2023
Bingshuo Chen, Xiaohua Zhao, Yang Li, Jianguo Gong, Xiaoming Liu
A total of 20 younger and 18 older drivers volunteered to participate in this study. The ratio of male and female subjects (28 men and 10 women) is close to that of licensed drivers in China (Zhao et al., 2022). According to the central limit theorem, if the random variables are normally distributed, the large sample size obtained from these variables also conforms to the normal distribution. The rule of thumb commonly used in empirical studies on driving behavior involves no less than 30 drivers (Zhao et al., 2021). Therefore, the sample size of participants is considered reasonable. All the drivers are required to hold a valid driver’s license and have normal vision and hearing ability.