Finite population corrections

Another complication Is a finite population correction which takes the form of... 

The quantity [ N — n) JNl] is known as a finite population correction -factor. When the number Nl of particles in the lot is very large relative to the number n of particles in the sample, [(V — n)/Ni] 1. The Var(FE) will then be approximately equal to the statistical rel. var. (sample mean). 

The factor [(.V — n)/N], which can be expressed as [1 - (n/N)], reduces the magnitude of the variance of the mean by the sampling fraction when compared to an infinite population value. This reduction factor is called ihe finite population correction factor or fpc, and it indicates the improved quality of the information about the population when n is large relative to N. As n grows larger, the variance of the population mean decreases and becomes zero if n = N, since at this point the mean is known exactly. In many situations fpc has a minimal effect and is usually ignored if n/N < 0.05, and then [(A — //)/A ] is set equal to 1. The confidence interval at P = a, for the estimated mean y, is given by... 

Equation [21] was derived for the case when the sample is very small in comparison with the whole population. For the case when the sample is a substantial part of the population, a so-called finite population correction factor, (1—should be introduced. Here, N is the total number of samples contained in the whole population, the second term of the right-hand side of eqn [19] should be multiplied by this factor. Consequently, eqn [21] takes the following form ... 

Evidently, the finite population correction should be taken into consideration only for relatively small, finite population cases. It can be neglected whenever N is sufficiently large, as then (1 — approaches unity. Equations [21] and [22] can be used for estimating the number of samples necessary to confine the sampling error within +e with the confidence level chosen for the t value (95%, for example), provided the estimated variance within the sample unit is Sj. 

The above formula assumes that the sample size n is much smaller than the total population size N, so that population can be considered infinite in size. When the sampling fraction is large (approximately 10 % or more) the estimate of the error must be corrected by multiplying by a finite population correction (FPC) [2]... 

Now the reduction for the 99.99 % probability of survival has grown to 10 % from 241.7 MPa down to 216 MPa. As last, the designer wants to know how far or close these 140 samples are from the population size to use the finite population correction (FPC) given by Eq. 4.9. To that purpose he calculates the overall volume of all the work pieces that must be built with that steel to assess N. The overall volume is equal to 2.2 x 10 mm. Since the volume of the traction specimen used is equal to 7,854 mm the number N of specimens of the entire population is N = 2.2 x 10 /7.854 = 280,112. He concludes that 140 specimens represent only a mere 0.05 % of the entire population size and he cannot apply any correction. This simple example evidences the difference that may arise when using a sample of limited size. Statistics results depend on the sample size. We may try, then, to use the Lieberman one-sided tolerance limit. The mean value of the logarithms of experimental data is og au) = 2.717 and 5 = 0.02175 from Table 4.5 we infer that the value of k for a sample of size 140 and a probability of survival of 99.95 % is k = 3.356 therefore ... 

In the case of spray towers it has been shown by Thornton 10 that ur is well represented by Hod — j) where u0 is the velocity of a single droplet relative to the continuous phase, and is termed the droplet characteristic velocity. The term (1 — j) is a correction to m0 which takes into account the way in which the characteristic velocity is modified when there is a finite population of droplets present, as opposed to a single droplet. It must be seen therefore that for very dilute dispersions, that is as j -> 0, w0(l — j) o- On the other hand, as the fractional hold-up increases, the relative velocity of the dispersed phase decreases due to interactions between the droplets. Substituting for ur, equation 13.32 may be written as ... 

These approximations are reliable apart from when a(r) is very small, as Figure 18.13 confirms. As a(r) —> 0, the remaining mass approaches 1, which is the correct (but practically useless) limit for end-chain scission of a population with initially infinite degree of polymerization. For a realistic population with initially finite degree of polymerization, this is unhelpful as the correct limit is 1 - x/n. It may be shown that when 0(a(r)) > 1 In, and n is large, the initial distribution of polymer molecules has little effect on the frequencies and the approximations remain valid. 

When epidemiologists compare two human populations, one defined as being at risk and the other defined as the control, they begin by hypothesizing that there is no difference in disease frequency between the two populations. They then collect data to decide whether their hypothesis is correct or incorrect. The hypothesis of no difference between two populations is called the null hypothesis. The null hypothesis is accepted if it is decided that there is no difference between the two populations, and it is rejected if it is decided that there is a difference. There is a finite probability of committing an error and rejecting the null hypothesis when it should be accepted and of accepting the nnll hypothesis when it should be rejected. The decision to accept or reject the null hypothesis is associated with a specified level of statistical confidence in the data. For example, if the null hypothesis is rejected at the 0.95 confidence level, there is a 95% chance that the decision is correct (i.e., that there really is a difference between the study and control populations) and a 5% chance that the decision to reject the null hypothesis is erroneous (i.e., that there really is no difference between the two populations). 

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