Monitoring the Nonconforming Fraction with a Dynamic Scheme When Sample Sizes are Time-varying

In many practical applications, it is more convenient to characterize the quality of production processes or service operations throughout the count of nonconformities. In the context of SPC, nonconformities are usually assumed to appear according to the binomial probability model. The conventional way for monitoring nonconformities involves Shewhart-type control procedures based on both constant and time-varying sample sizes. In this article, an EWMA scheme is proposed for monitoring the fraction of nonconforming items with time-varying sample sizes. The proposed control chart is referred to as the EWMAG-B and can be easily adapted to work with a constant sample size by fixing it at a needed value. By means of simulation, it was found out that the EWMAG-B chart outperforms the conventional p control chart in Phase II while detecting changes in the process level is wanted.

Genetic Variation

An introduction to randomness and the neutral theory of molecular evolution and how it contributes to levels of genetic diversity in populations. The estimation of random shifts in allele frequencies over time is discussed using the binomial probability distribution. This chapter visualizes how different initial allele frequencies relate to their probability of being fixed in a population when only drift is acting. The outputs of multiple simulations of a population undergoing random increase and decrease in allele frequencies per generation are built, run, and visualized. The fixation index (F) is introduced to quantify the loss of genetic variance over time.

ISLAMIC MUTUAL FUNDS PROMOTIONAL ACTIVITIES: NORMAL APPROXIMATION OF BINOMIAL DISTRIBUTION AND TYPE I ERROR

The paper applies normal approximation procedure to binomial probability distribution. A sample of 392 respondents are surveyed whether they agree or not agree that promotional activities determined the level of awareness of benefits of Islamic mutual funds. The paper hypothesizes the population mean µ of success effects of promotional activities at 68%, and attempts to reduce Type I error - the probability of rejecting null hypothesis when it is true.

A Family of High Continuity Subdivision Schemes Based on Probability Distribution

Subdivision schemes are famous for the generation of smooth curves and surfaces in CAGD (Computer Aided Geometric Design). The continuity is an important property of subdivision schemes. Subdivision schemes having high continuity are always required for geometric modeling. Probability distribution is the branch of statistics which is used to find the probability of an event. We use probability distribution in the field of subdivision schemes. In this paper, a simplest way is introduced to increase the continuity of subdivision schemes. A family of binary approximating subdivision schemes with probability parameter p is constructed by using binomial probability generating function. We have derived some family members and analyzed the important properties such as continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils. It is observed that, when the probability parameter p = 1/2, the family of subdivision schemes have maximum continuity, generation degree and Holder regularity. Comparison shows that our proposed family has high continuity as compare to the existing subdivision schemes. The proposed family also preserves the shape preserving property such as convexity preservation. Subdivision schemes give negatively skewed, normal and positively skewed behavior on convex data due to the probability parameter. Visual performances of the family are also presented.

Estimating confidence intervals for gravel bed surface grain size distributions

Most studies of gravel bed rivers present at least one bed surface grain size distribution, but there is almost never any information provided about the uncertainty of the percentile estimates. We present a simple method for estimating the confidence intervals about the grain size percentiles derived from standard Wolman or pebble count samples of bed surface texture. Our approach uses binomial probability theory to generate confidence intervals for all grain sizes in the distribution. We find that the standard sample size of 100 observations is associated with errors ranging from about ±15 % to ±30 %, which may be unacceptably large for many applications. In comparison, a sample of 500 stones produces an uncertainty ranging from about ±9 % to ±18 %. In order to help workers develop appropriate sampling approaches that produce the desired level of precision, we present simple equations that approximate the proportional uncertainty associated with the median size and the 84th percentile of the distribution as a function of the sample size and the standard deviation of the distribution, assuming that the underlying distribution is log-normal. However, the true uncertainty of any sample can only be accurately estimated once the sample has been collected, so these simple equations complement – but do not replace – the basic uncertainty analysis using binomial probability theory.