Influence of Sample Size

Once the distribution function for the design property has been selected, the next step is to establish a design allowable. Such a value is based on the determina-tion/estimation of some specific design property and its variability. For reliability analyses, the requisite property, or properties, must be quantified and statistically 

Traditionally, safety factor is defined as a knock down factor to reduce the allowable design stress to a lower level than the cogent property (say, the minimum tensile strength) of the material. Recognizing the fact that such a property can vary from point to point within a product form, and from lot to lot, such a property is grouped and represented through an appropriate statistical distribution, and characterized through the appropriate statistical parameters. If the population mean and variance (/x and a) are known, the lower bound value of the property X (namely, X) can be defined as X = p, - fia, the value of p is chosen to provide a measure that the probability that X will be below (or above) X. A safety factor S.F.) may then be defined as  

For example, for = 3.0, the S.F. would provide a probability of failure of 1.35 x 10 or 0.135%, or a 99.86% probability of survival, which are well defined (assuming, of course, that the property is appropriately represented by the distribution). But, because the mean p. and standard deviation a are both not known, safety factors must be defined in terms of their estimates based on n measurements (i.e., on A( ) and 5( )), with some defined confidence level for the estimates, where both the population mean (p.) and the standard deviation ( t) would be estimated from the limited measured data. 

As a numerical illustration, it will be assumed that the standard deviation a of the Normal distribution is known, and the mean p of the distribution is to be estimated from the sample average, say X( ), from n measurements. The sample average A( ) is a random variable with a standard deviation equal to a/.Jn. The mean p) of the distribution is estimated by Eqn. 5.13 and is given by  

Note that, in practice, the mean and variance of the distribution (/u. and a) are not precisely known, and are estimated from n measurements. As such the lower-bound estimate of the mean, at a prescribed confidence level, used to represent the mean of the distribution, and some multiple of the standard deviation (or standard error) is used to define the design allowable, namely, 

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Hypothetical member of underlying population Figure 14.4 Influence of sample size on the outcome of significance testing... 

Fig. 7.8. Influence of sample size on the recovery of (A) PCB-52 from river sediment ( 100 mg, I g) and (B) cocaine from coca leaves ( 220-470 pg, 150-170 pg, A less than 150 pg). (Reproduced with permission of Springer International and Elsevier, respectively.)...

Experimental Study of System Peak Profiles. Influence of Sample Size... 

X is associated with some probability for failure, which depends on the choice of the distribution function. For illustration, (a) the influence in choice of the distribution function is considered in terms of the Normal (or Gaussian) and the Weibull distributions, and (b) the influence of sample size is examined using the Normal distribution. 

The influences of sample size and material quality (or property variability) are illustrated in Table 5.3. The property is identified with plane strain fracture toughness (i.e., X( ) = Kjc( ) = 50ksi Vim, with a = 2.5 or 5.0 ksi Vim to reflect two different levels of variability, or manufacturing control). The results show marginal improvements with increasing number of tests to characterize variability in the property data, and substantial improvement in variabihty with quality control. It is recognized that the values shown at the lower numbers of test samples (say, n = 5) are inappropriate, but they do convey the need for a quantitative basis for design and reliability analyses. 

W. Harrer, R, Danzer, P. Supancic, and T. Lube, Influence of sample size on the results of B3B tests. Key Engineering Materials, in print, (2009). 

Variance is a function of the size of individual samples and sample size is therefore of considerable importance in the analysis of mixture quality. All of the above indices are therefore dependent on sample size and this dependence has been investigated . Similarly, the influence of sample size on the estimation of interface length can be made from mixture patterns. 

Some initial analyses of auto-correlation and related assessments of mixing have been conducted and given the capability and low cost of modem image analysis equipment these methods offer a basis for detailed, fine scale analysis of mixture patterns. It is important to re-emphasize the influence of sample size or scanning spot size on such analyses. Ideally, this size should be as small as the smallest sub-division of the best mixed cross-section to be analysed but, failing that, size needs to be standardized for any comparative work. 

Present use of the plots of ojC and L,/D, assume that there is no dfect of viscosity or flow rate on mixing rate, only on pressure drop. Similarly it is often assumed that the use of L,/D, takes account of scale. It also seems worthwhile to look at the more detailed analysis of mixture cross-section by autocorrelation and related indices as this provides a more precise description of the state of mixedness than the determination of variance and also a possibility of exploring the influence of sample size. Other topics needing further investigation are the influence of scale and non-Newtonian fluid properties - - ... 

In the last section, we considered the question of sample size as it relates to significance testing. A model discussed by Trenholm etal. (1979) provides an example of the influence of sample size in the context of estimation. 

The quench test was performed on two of the sintered bars, as representative samples, designated as Samplel Sample3 , (10 0.13 mm X 10 0.13 mm in cross section). The notation Samplel Sample3 corresponds to two bars having a difference in geometry, primarily thickness ail other processing parameters being exactly the same for both. The thickness was varied in order to study the influence of sample size on the response to thermal shock treatment. 

Crapper McLachlan, D. R., Krishnan, S. S., Quittkat, S. and DeBoni, U. (1980) Brain aluminum in Alzheimer disease influence of sample size and case selection. Neuroto dcol. 1, 25-32. 

Figure 8. (A) Influence of sample size on the heat output per cell in a suspension of human renal carcinoma cells. Note that it decreases with increasing cell number. This may be expressed by a continuous transition from aerobiosis in a monolayer-like suspension to increasingly anaerobic conditions in a crowded ampoule. (B) Influence of sample size on heat output per unit dry weight (dw) in rat liver tissue samples. It increases with decreasing sample size. This is apparently due to a continuous transition from mainly anaerobic conditions in large tissue slices to an increasing amount of aerobiosis in biopsy -Iike samples. The two curves complement each other giving a sigmoidal relationship, whose steep portion fits a linear regression when plotted on log scales (Reproduced from Reference [43] with permission).

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